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NEW 


COMMERCIAL  ARITHMETIC 


BY 
JOHN   H.    MOORE 


NEW  YORK-:.  CINCINNATI-:.  CHIC  AGO 

AMERICAN    BOOK    COMPANY 


COPYRIGHT,   1904  AND   1907,    BY 

L.  L.  WILLIAMS  AND  F.  E.  ROGERS. 
ENTERED  AT  STATIONERS'  HALL,  LONDON. 

MOORE'S  COM.  AB. 

EDUCJTTON  DEFT* 


PREFACE 

A  commercial  arithmetic  should  be  comprehensive  in  its  scope, 
but  should  contain  no  complicated  or  obsolete  subjects.  It  should 
furnish  abundant  material  for  drills  in  modern  business  problems, 
and,  by  natural  and  progressive  steps  in  the  methods  of  developing 
the  subjects  presented,  should  cultivate  in  the  student  those  qualities 
of  accuracy,  rapidity,  and  self-reliance  that  will  be  so  valuable  to 
him  later. 

With  these  objects  in  mind  this  book  has  been  written.  It  is 
not  intended  for  beginners,  but  for  students  pursuing  a  commercial 
course  in  business  and  secondary  schools.  While  it  may  be  assumed 
that  these  students  have  previously  completed  a  more  elementary 
arithmetic,  yet  experience  has  demonstrated  that  it  is  usually  neces- 
sary for  them  to  review  the  fundamental  operations,  and  become 
familiar  with  the  short  methods  which-  are  applicable  to  simple  calcu- 
lations, before  they  can  do  effective  work  in  commercial  arithmetic. 
The  underlying  principles  of  arithmetic  are,  therefore,  briefly  re- 
viewed, and  many  practical  counting-room  methods  having  a  direct 
bearing  upon  them  are  carefully  illustrated  and  explained. 

Great  care  has  been  taken  to  make  the  methods  of  developing  all 
the  principles  natural  and  businesslike.  All  of  the  operations  given 
in  connection  with  the  illustrative  problems  are  accompanied  with 
solutions  which  enable  the  student  to  understand  the  principles 
involved.  The  student  is  taught  to  understand  a  process  before  he 
is  taught  to  summarize  it  in  a  rule.  Solutions  and  rules  are  omitted 
in  all  cases  where  it  is  thought  the  student  can  prepare  them  without 
assistance.  The  few  rules  given  in  the  book  all  follow  solutions,  and 
are  intended  to  aid  the  student  to  produce  intelligent  results.  In 
no  case  are  they  intended  to  be  committed  to  memory. 

Mental  work  has  received  due  emphasis  throughout  the  book. 
Oral  exercises  of  a  thoroughly  practical  nature  accompany  every  sub- 
ject, and  in  many  cases  methods  of  computation  are  introduced  and 
developed  through  a  series  of  oral  drills. 

3 


4  PREFACE 

An  attempt  has  been  made  to  make  the  treatment  of  the  whole 
subject  highly  educative,  but  methods  and  topics  distinctively  utili- 
tarian in  their  value  have  received  due  attention.  Arithmetical 
puzzles  and  improbable  conditions  have  been  studiously  avoided, 
and  a  feature  is  made  of  concrete  business  problems  from  the  outset. 
Particular  attention  has  been  devoted  to  the  subject  of  Addition. 
The  group  method  is  carefully  developed  through  a  series  of  oral 
and  written  drills.  The  exercises  on  tabulation  and  all  the  exercises 
calling  for  vertical  and  horizontal  additions  are  especially  valuable. 

Only  small  common  fractions  are  introduced ;  they  are  the  only 
ones  used  in  ordinary  business.  In  connection  with  this  subject 
special  care  has  been  devoted  to  the  topics  Quantity,  Price,  and  Cost, 
and  Bills  and  Accounts.  The  methods  developed  and  the  forms 
illustrated  in  this  part  of  the  book  are  especially  helpful  and  practi- 
cal. In  the  chapter  on  Denominate  Numbers  a  feature  is  made  of  the 
subject  Practical  Measurements.  In  the  preparation  of  this  portion 
of  the  book  the  author  consulted  mechanics,  contractors,  and  busi- 
ness men, 'thoroughly  versed  in  their  several  departments,  in  order 
to  get  at  current,  practical  usages.  In  the  chapter  on  Percentage 
and  its  Applications,  the  subjects  Commercial  Discounts,  Interest, 
Bank  Discount,  and  Customhouse  Business  have  been  especially 
emphasized  because  they  are  so  closely  connected  with  modern  busi- 
ness transactions.  In  the  chapter  on  Sharing,  the  subject  Partner- 
ship has  been  thoroughly  covered.  All  the  problems  given  in  this 
work  are  treated  from  the  accountant's  standpoint,  and  are  entirely 
free  from  all  unusual  conditions.  In  the  preparation  of  all  the  sub- 
jects, business  men  have  been  consulted  freely. 

In  connection  with  many  of  the  subjects  a  great  deal  of  valuable 
information  is  given.  Numerous  business  forms  are  also  introduced, 
and  made  the  basis  of  a  series  of  problems. 

Some  of  the  problems  given  have  been  taken  from  the  Williams 
and  Bogers's  Commercial  Arithmetic,  by  Oscar  F.  Williams ;  but  the 
majority  of  them  are  new. 

Acknowledgment  is  due  to  Professor  C.  D.  Clarkson  of  the  Depart- 
ment of  Commerce  in  Drexel  Institute,  Philadelphia,  for  valuable 
assistance  in  perfecting  the  volume. 


CONTENTS 

SIMPLE  NUMBERS  PAGE 

Preliminary  Definitions .7 

Notation  and  Numeration .  8 

Addition            12 

Subtraction 25 

Multiplication 36 

Division             47 

Properties  of  Numbers .50 

UNITED  STATES  MONEY 58 

METHODS  FOR  PROVING  WORK 65 

FRACTIONS 

Common  Fractions 70 

Decimal  Fractions 93 

Quantity,  Price,  and  Cost          .        .        .        .        .        .        .        .  108 

Bills  and  Accounts    ..........  121 

DENOMINATE  NUMBERS 

Measures 134 

Denominate  Quantities 150 

Practical  Measurements 163 

PERCENTAGE  AND  ITS  APPLICATIONS 

Percentage ' 184 

Commercial  Discounts      .........  198 

Gain  and  Loss 206 

Marking  Goods 214 

Commission 219 

Interest              . 228 

Present  Worth  and  True  Discount 256 

Negotiable  Paper 259 

Bank  Discount 263 

Partial  Payments 272 

Equation  of  Accounts 277 

Cash  Balance 291 

Savings-bank  Accounts 296 

Stocks 300 

Bonds 310 

5 


6"  CONTENTS 

PERCENTAGE  AND  ITS  APPLICATIONS  PAGE 

Insurance 314 

Taxes 326 

Customhouse  Business 333 

Exchange           . 340 

SHARING 

Proportional  Parts 357 

Partnership 359 

Building  and  Loan  Associations       .......  371 

RATIO  AND  PROPORTION 

Ratio 377 

Proportion 378 

STORAGE 

Cash  Storage 381 

Credit  or  Average  Storage 383 

APPENDIX 

Metric  System  of  Measures .         .  385 

Powers  and  Roots 389 

Compound  Interest  Table  for  Annual  Payments     ,        .        .        „  398 


NEW   COMMERCIAL   ARITHMETIC 


SIMPLE  NUMBERS 

PRELIMINARY  DEFINITIONS 

1.  Arithmetic  is  the  science  of  numbers  and  the  art  of  com- 
puting by  them. 

2.  A  unit  is  a  single  thing,  or  a  definite  quantity  regarded  as  a 
single  thing. 

In  selling  cloth  by  the  yard  the  unit  is  one  yard  of  cloth ;  in  measuring  lands 
by  the  acre  the  unit  is  one  acre  of  land ;  in  counting  the  number  of  students  in 
a  class  the  unit  is  one  student ;  in  buying  bricks  by  the  thousand  the  unit  is  one 
thousand  bricks  ;  in  selling  "posts  by  the  hundred  the  unit  is  one  hundred  posts. 

3.  An  integral  unit  is  one,  or  a  whole  thing. 

4.  A  decimal  unit  is  one  of  the  parts  obtained  by  dividing  an 
integral  unit  into  tenths,  hundredths,  and  so  on. 

5.  A  fractional  unit  is  one  of  the  parts  obtained  by  dividing  an 
integral  unit  into  any  number  of  equal  parts. 

6.  A  number  is  a  unit  or  two  or  more  units. 

7.  An  integer  is  an  integral  unit  or  two  or  more  integral  units. 

8.  An  abstract  number  is  a  number  not  associated   with   any 
particular  thing  or  quantity;    as,  2,  7,  11. 

9.  A  concrete  number  is  a  number  associated  with  some  particu- 
lar thing  or  quantity ;  as,  11  men,  6  cords  of  wood. 

10.  A    denominate  number    is    a    concrete    number    expressing 
standard  money  value,  or  standard  measure  or  weight;  as,  1  dollar; 
5  gallons ;  6  pounds,  4  ounces. 

11.  Like  numbers  are  numbers  that  have  the  same  unit  value  ;  as, 
2,  6,  9 ;  3  houses,  7  houses,  5  houses ;  2  years,  9  years,  20  years. 


•r. 
I    • 

"S        '  '•  SIMPLE  NUMBERS  [§§  12-21 


'121  TJnKke  numbers  are  numbers  that  have  different  unit  values ; 
as,  12,  16  days,  4  boys,  2  hours. 

13.  A  simple  number  is  a  number  consisting  of  a  unit  or  a  collec- 
tion of  units  of  the  same  kind ;  as  2,  12  men,  8  pounds. 

14.  A  compound  number  is  a  number  consisting  of  two  or  more 
denominations  of  the  same  unit;  as,  7  bushels,  3  pecks,  1  quart; 
6  pounds,  3  ounces. 

15.  A  problem  is  a  question  to  be  solved. 

16.  A  principle  is  a  general  law  used  as  a  basis  for  computations. 

17.  A  rule  is  a  concise  outline  of  the  steps  to  be  taken  in  the 
performance  of  a  computation. 

ORAL  EXERCISE 

1.  All  denominate   numbers   are  concrete.     Are   all   concrete 
numbers  denominate  ?     Explain. 

2.  All  the  following  numbers  are  concrete.     Are  they  denomi- 
nate ?    Explain.     16  pounds,  12  men,  4  rods,  7  dollars,  9  houses. 

8.  State  clearly  the  difference  between  a  concrete  number  and  a 
denominate  number. 

4-   Give  an  example  of  a  compound  number. 

d.  Is  there  any  difference  between  a  compound  number  and  a 
denominate  number?  Explain. 

6.  What  is  the  unit  of  16?     of  75  barrels  of  molasses?    of 
$7500?    of  ^  of  a  week?    of  2\  dozen? 

7.  Give  an  example  of  a  simple  abstract  number ;  of  a  simple 
concrete  number. 

8.  Name  two  like  numbers ;  two  unlike  numbers. 

NOTATION  AND  NUMERATION 

18.  Notation  is  the  art  of  writing  numbers. 

19.  Numbers  are  generally  expressed  by  figures  or  letters,  but 
they  may  also  be  expressed  by  words. 

20.  Numeration  is  the  art  of  giving  oral  expression  to  numbers. 

21.  The  two  methods  of  notation  in  use  are  the  Arabic  and  the 


§§22-26]  NOTATION  AND  NUMERATION  9 

ARABIC  NOTATION 

22.  The  Arabic  Method  of  Notation,  first  used  by  the  Arabs,  com- 
prises ten  characters  or  figures,  as  follows : 

O  123J4567  8  9 

Naught     One     Two     Three      Four     Five        Six      Seven       Eight       Nine 

The  figures  1,  2,  3,  4,  6,  6,  7,  8,  9  are  called  digits,  and  the  figure  0  is  called 
eero,  naught,  or  cipher. 

23.  The  value  of  a  digit  is  determined  (1)  by  its  name,  and  (2)  by 
its  position  in  a  number.      A  digit,  when  standing  alone,  always 
equals  the  number  of  units  which  its  name  indicates;  when  com- 
bined with  other  digits  its  value  is  determined  by  the  place  which 
it  occupies  in  the  number. 

24.  The  value  of  a  digit  in  any  given  number  increases  from 
right  to  left,  and  decreases  from  left  to  right  in  a  tenfold  ratio. 

Thus,  in  the  number  eleven,  expressed  11,  the  second  1  from  the  right  has  a 
value  ten  times  as  great  as  the  first  1. 

25.  Orders  of  Units.     The  place  which  a  figure  occupies  in  a 
number  is  called  its  order.     The  ones  of  a  number  are  called  units 
of  the  first  order;  the  tens,  units  of  the  second  order;  the  hundreds, 
units  of  the  third  order;  the  thousands,  units  of  the  fourth  order;  and 
so  on. 

Ten  units  of  any  given  order  are  equal  to  one  unit  of  the  next 
higher  order. 

26c  Periods.  Numbers  containing  four  figures  or  more  are,  for 
convenience,  separated  by  the  comma  into  periods  of  three  figures 
each.  Beginning  at  the  right,  the  first  group  is  the  period  of  units; 
the  second,  the  period  of  thousands;  the  third,  the  period  of  millions; 
the  fourth,  the  period  of  billions ;  and  so  on. 

One  thousand  units  of  any  given  period  are  equal  to  one  unit  of  the 
next  higher  period. 

The  left-hand  period  of  any  number  may  consist  of  one,  two,  or 
three  figures. 


10 


SIMPLE   NUMBERS 


[§2d 


NUMERATION   TABLE 


1 

PERIOD 

NUMBER 

NUMERATION 

ORDER 

H* 

Hundreds  of  quintillions 

21st 

7th 

Oi 

Tens  of  quintillions 

20th 

CO 

Quintillions 

19th 

0) 

Hundreds  of  quadrillions 

18th 

6th 

-5 

Tens  of  quadrillions 

17th 

CD 

Quadrillions 

16th 

H-  1 

Hundreds  of  trillions 

15th 

5th 

0> 

Tens  of  trillions 

14th 

M 

Trillions 

13th 

0 

Hundreds  of  billions 

12th 

4th 

0& 

Tens  of  billions 

llth 

CO 

Billions 

10th 

CO 

Hundreds  of  millions 

9th 

3d 

M 

Tens  of  millions 

8th 

~J 

Millions 

7th 

00 

Hundreds  of  thousands 

6th 

2d 

CO 

Tens  of  thousands 

5th 

Oi 

Thousands 

4th 

Hundreds 

3d 

1st 

CO 

Tens 

2d 

^ 

Units 

1st 

ORAL  EXERCISE 

1.  Read  the  following  numbers:   5,005;   1,925;   3,036;   4,569; 
260;  715. 

NOTE.  In  reading  numbers,  always  express  them  in  the  shortest  way 
possible.  Thus,  1520  should  be  read  fifteen  hundred  twenty,  not  one  thousand 
five  hundred  twenty.  This  is  important  in  writing  amounts  in  checks,  notes, 
and  drafts,  where  the  space  is  often  limited. 

Do  not  read  and  between  periods  or  between  hundreds  and  units.  Thus, 
16,725  should  be  read  sixteen  thousand,  seven  hundred  twenty-five,  not  sixteen 
thousand  and  seven  hundred  and  twenty-five.  This  distinction  is  of  the  utmost 
importance  in  connection  with  the  writing  of  decimals. 

2.  What  is  the  name  of  the  second  period  of   notation?    the 
third  ?   the  fourth  ?   the  fifth  ?   the  seventh  ? 


§§26-29]  NOTATION  AND  NUMERATION  11 

8.   What  are  the  names  of  the  successive  periods  expressed  by 
seven  figures  ?   by  eleven  figures  ? 

4-   How  many  figures  are  required  to  write  millions  ?  trillions  ? 

5.  How  many  units  of  the  first  order  in  the  second  period 
of  9,321  ? 

6.  Eead  the  following  numbers:  246,920,460;  750,861,432,120; 
9,246,921,006 ;  1,269,247,268,490,621 ;  700,600,070,000,000,002. 

7.  Express  in  figures  three  units  of  the  fourth  order,  three  of 
the  third,  nine  of  the  second,  and  seven  of  the  first. 

8.  Distinguish  between  an  order  and  a  period  as  related  to 
numbers. 

ROMAN  NOTATION 

27.  The  Roman  Method  of  Notation  is  used  extensively  in  number- 
ing volumes,  chapters,  sections,  and  the  other  important  divisions  of 
books;  also  in  numbering  dials  and  tabular  outlines.     It  employs 
seven  characters  or  letters,  as  follows : 

I          V  X  L  O  D          M 

1  5  1O          50        100       500      100O 

28.  Roman  Values.     The  value  of  Roman  characters  is  twofold. 

1.  Each  character  when  standing  alone  has  a  definite  value,  as 
above. 

2.  Each  character  also  has  a  varying  value  when  written  in 
varying  positions  in  combination  with  other  Roman  numerals. 

29.  General  Principles.    1.  Repeating  a  letter  repeats  its  value. 
Thus,  II  represents  two  ;  XX,  twenty;  CCC,  three  hundred. 

2.  When  a  letter  of  less  value  is  placed  before  one  of  greater 
value  the  number  indicated  is  the  difference  between  the  values  of 
such  numbers. 

Thus,  IX  represents  nine  ;  XC,  ninety. 

3.  When  a  letter  of  less  value  is  placed  after  one  of  greater  value, 
the  number  indicated  is  the  sum  of  the  values  of  such  letters. 

Thus,  CX  represents  one  hundred  ten;  LXIV,  sixty-four. 


12 


SIMPLE   NUMBERS 


[§§  29-34 


4.  A  bar  placed  over  a  letter  multiplies  the  value  of  the  letter  by 
one  thousand. 

Thus,  V  represents  five  thousand  ;  C,  one  hundred  thousand. 

5.  A  letter  should  not  be  repeated  more  than  three  times  in 
expressing  numbers. 

6.  A  bar  is  never  placed  over  the  letter  I. 

TABLE  OF  ROMAN  NUMERALS  WITH  ARABIC  EQUIVALENTS 


I    ....    1 

XII     ...  12 

L  ....     50 

DCC  .     .          700 

II    ....     2 

XIII    ...  13 

LX    .     .     .     60 

DCCC    .          800 

III.     ...     3 

XIV    ...  14 

LXX      .     .     70 

CM    .    .          900 

IV.     ...     4 

XV     ...   15 

LXXX  .     .     80 

M.     .     .         1000 

V    ....     5 

XVI    ...  16 

XC    .     .     .     90 

MM  .     .         2000 

VI.     ...     6 

XVII  .     .     .17 

C  .     .     .     .100 

V.     .     .         5000 

VII     ...     7 

XVIII      .     .  18 

CO     ...  200 

X.     .     .       10000 

VIII    ...     8 

XIX    ...  19 

CCC  ...  300 

L  .     .     .       50000 

IX.     ...    9 

XX      ...  20 

CD    ...  400 

C  .     .     .     100000 

X    .     .     .     .10 

XXX  ...  30 

D  .     .     .     .600 

D  .     .     .     500000 

XI  ....  11 

XL      ...  40 

DC    .     .     .600 

M.     .     .  1000000 

ORAL  EXERCISE 
Bead  the  following  expressions : 

XCII;   XXVII;   XXIX;   CCXVII;   DLXX;  DCC;   MDCCCLIII; 
MMDXLIV;     MCDLXX. 

ADDITION 

30.  Addition  is  the  process  of  combining  several  numbers  into 
one  equivalent  number. 

31.  The  sum  or  amount  is  the  result  obtained  by  addition. 

32.  The  sign  +  signifies  addition  and  is  read  plus. 

33.  The  sign  =  signifies  equality  and  is  read  equals. 

34.  General  Principles.     1.  Only  the  same  orders  of  units  of  like 
numbers  can  be  added. 

2.   The  sum  always  expresses  units  of  the  same  name  as  the 
several  numbers  to  be  added. 


<?§  34-37]  ADDITION  13 

3.   The  sum  of  two  or  more  numbers  is  the  same  in  whatever 
order  the  numbers  may  be  added. 

ORAL  EXERCISE 

1.  Beginning  at  7  count  by  7's  to  98  ;  by  6's  to  79. 

-  2.  Beginning  at  31  count  by  9's  to  112  ;  by  8's  to  95. 

8.  Beginning  at  17  count  by  4's  to  117  ;  by  12's  to  77 

4.  Beginning  at  49  count  by  6's  to  139  ;  by  8's  to  113. 

5.  Beginning  at  29  count  by  8's  to  93;  by  7's  to  78. 

6.  Beginning  at  72  count  by  5's  to  117  ;  by  9's  to  108. 

7.  Beginning  at  0  count  by  13's  to  156  ;  by  ll's  to  121. 

8.  Beginning  at  0  count  by  14's  to  126  ;  by  12's  to  108. 

9.  Beginning  at  0  count  by  15's  to  135  ;  by  17's  to  153. 

10.  Beginning  at  0  count  by  16's  to  144  ;  by  9's  to  144. 

11.  Beginning  at  29  count  by  15's  to  104  ;  by  9's  to  110. 

12.  Beginning  at  37  count  by  ll's  to  136;  by  7's  to  86. 

13.  Beginning  at  3  count  by  12's  to  147  ;  by  19's  to  60. 

14.  Beginning  at  4  count  by  18's  to  94  ;  by  17's  to  106. 

35.   Example.     Find  the  sum  of  945,  626,  924,  and  726. 

SOLUTION.     Since  only  units  of  the  same  order  can  be  added, 


write  units  under  units,  tens  under  tens,  and  hundreds  under  hun- 
dreds, and  draw  a  line  beneath.     Beginning  at  the  right-hand,  or 
"24       units'  column,  and  adding  downwards,  the  sum  is  21  units,  or  2  tens 
726        and  1  unit.     Write  1  in  the  units'  column  and  add  2  to  the  tens' 
3221       column,  obtaining  as  a  result  12  tens,  or  1  hundred  and  2  tens. 
Write  2  in  the  tens'  column  and  add  1  to  the  hundreds'  column, 
obtaining  32.     Write  this  entire  result  to  the  left  of  the  numbers  before  writ- 
ten.    The  required  result  is  3221. 

36.  To   insure    accuracy   in    addition   all    figures    should    be: 
(1)  uniformly  spaced;    (2)  legibly  written;   (3)  of  a  uniform  size. 

RAPID  ADDITION 

37.  The  secret  of  rapid  addition  lies   mainly  in  the  ability  to 
group  series  of  figures  with   facility.     In  reading  the  words  of  a 
sentence  we  do  not  look  at  the  individual   letters,  but  rather  at 
groups  of  letters  which  make  words;  so  in  attempting  to  add  col- 


14  SIMPLE   NUMBERS  [§§  37-39 

umns  rapidly  we  should  not  think  of  the  individual  figures,  but  of 
the  results  of  groups  of  figures. 

Thus,  in  adding  6,  9,  2,  3,  1,  2,  and  6,  we  should  not  say  6  and  9  are  15  and  2 
are  17  and  3  are  20  and  1  are  21  and  2  are  23  and  5  are  28;  but  15  (6  -1-  9), 
21  (15  +  2  +  3+1),  28  (21  +  2~+~5). 

38.  The  ability  to  group  numbers  readily  may  be  acquired  by 
intelligent,   persistent   practice.      By  constantly   aiming    to    form 
groups  in  adding,  we  gradually  become  master  of  a  vocabulary  of 
groups  which  will  eventually  serve  us  to  advantage  in  all  numerical 
operations  which  we  may  perform.    We  should  commence  by  group- 
ing two  figures,  then  three,  and  so  on  until  we  can  take  in  at  a  glance 
from  two  to  four  figures  in  all  work. 

39.  Addition  is  one  of  the  most  important  and  one  of  the  most 
frequently  used  operations  of  arithmetic.     It  is  the  key  to  all  rapid 
business  calculations,  and  should  be  thoroughly  mastered  before  any 
of  the  other  principles  of  commercial  arithmetic  are  attempted. 

ORAL  EXERCISE 

In  the  following  exercise  the  student  should  make  combinations 
or  groups  of  two  figures  in  finding  the,  totals.  The  work  should  be 
done  rapidly,  the  student  speaking  aloud  the  successive  results. 

Thus,  in  problem  1,  below,  beginning  at  the  top  and  adding  downwards, 
results  should  be  named  as  follows:  8,  18,  24,  30,  9,  49.  Only  the  unit  figure 
should  be  pronounced  when  the  amount  continues  in  the  same  tens. 

1.          2.          8.  4.  5.  6.          7.          8.          9.       10. 


2 

3 

2 

4 

7 

3 

2 

7 

3 

9 

6 

2 

1 

3 

2 

2 

1 

8 

2 

2 

7 

1 

7 

1 

1 

7 

7 

2 

4 

4 

3 

2 

2 

2 

4 

9 

2 

4 

2 

5 

1 

6 

.  9 

3 

6 

2 

6 

6 

3 

1 

5 

7 

3 

6 

3 

1 

3 

2 

6 

2 

4 

3 

1 

2 

5 

4 

4 

7 

2 

6 

2 

2 

6 

1 

1 

8 

1 

2 

5 

3 

8 

4 

3 

5 

6 

3 

5 

8 

5 

5 

1 

6 

5 

3 

4 

3 

7 

2 

4 

5 

5 

2 

2 

2 

7 

1 

2 

3 

5 

7 

5 

7 

1 

5 

2 

2 

5 

7 

3 

4 

I  3S]  ADDITION  15 

11.        12.         13.         14.         15.         16.         17.         18          19.       20. 


3 

6 

3 

3 

3 

2 

3 

2 

2 

7 

7 

2 

2 

4 

2 

8 

2 

4 

5 

4 

2 

4 

8 

7 

8 

1 

1 

3 

4 

3 

5 

3 

6 

6 

6 

7 

4 

1 

1 

5 

3 

2 

1 

2 

1 

2 

6 

7 

7 

6 

2 

1 

2 

5 

2 

4 

2 

4 

2 

7 

1 

7 

7 

8 

7 

5 

8 

9 

9 

3 

7 

2 

5 

3 

5 

4 

4 

3 

3 

5 

3 

8 

4 

7 

4 

6 

5 

7 

7 

9 

2 

4 

5 

3 

6 

4 

3 

9 

3 

2 

7 

2 

7 

4 

5 

8 

7 

4 

1 

9 

3 

1 

5 

6 

5 

2 

3 

6 

9 

5 

Drill  on  the  foregoing  and  similar  combinations  until  you  can  make  groups 
of  two  figures  each  and  combine  them  in  a  total  as  rapidly  as  you  can  count  1,  2, 
3,  etc.  Next  use  the  foregoing  and  similar  exercises  in  drilling  upon  adding  by 
groups  of  three  figures  each. 

WRITTEN  EXERCISE 

Copy  or  write  from  dictation  and  find  the  sum  of: 

1.       2.  3.  4.      5.       6.  7. 

8481  4615  4521  3146  2610  1652  1431 

2341  9184  6210  7214  3115  1748  2115 

4678  8632  1940  1431  4221  2631  6211 

3444  1531  7249  1625  1635  4217  2542 

1234  3116  2614  3126  1724  2724  1625 

5678  4227  1837  1847  1142  1925  1143 

9212  1328  9246  2932  2416  1839  2748 

3456  2014  2143  1621  1345  4114  1932 

3231  9126  3214  4217  1621  1028  1647 

1645  3214  9125  2114  1942  1686  4212 

NOTE.  Any  of  the  above  or  similar  problems  may  be  copied  on  the  board 
and  each  student  in  turn  required  to  add  aloud,  making  groups  of  from  two  to 
four  figures.  The  student  should  begin  with  groups  of  two  figures  and  gradually 
work  up  to  groups  of  three  or  four  figures.  He  should  be  required  to  speak 
results  only.  Thus,  in  problem  7,  grouping  two  figures,  he  should  say,  6,  9,  17, 
27,  36,  in  adding  the  first  column ;  7,  12,  18,  25,  30,  in  adding  the  second  col- 
umn :  8,  15,  22,  38,  46,  in  adding  the  third  column  ;  7,  15,  17,  20,  25,  in  adding 
the  fourth  column.  The  rate  of  naming  the  successive  results  may  be  slow  at 
first,  but  it  should  be  gradually  quickened  as  facility  is  attained. 


16  SIMPLE   NUMBERS  [§  3P 

ORAL  EXERCISE 

1.  Beginning  at  37  count  by  8's  to  77 ;  by  9's  to  73. 

2.  Beginning  at  19  count  by  7's  to  47 ;  by  8's  to  51. 

8.  Beginning  at  29  count  by  12's  to  77  ;  by  14's  to  71. 

4.  Beginning  at  26  count  by  15's  to  86 ;  by  9's  to  62. 

5.  Beginning  at  37  count  by  8's  to  101 ;  by  9's  to  118. 

6.  Beginning  at  13  count  by  8's  to  173 ;  by  7's  to  160. 

7.  Beginning  at  14  count  by  13's  to  66 ;  by  14's  to  84. 

8.  Beginning  at  52  count  by  9's  to  133;  by  16's  to  180. 

DRILL  EXERCISE 
Pronounce  at  sight  the  totals  of  the  following  combinations : 


3 
5 

7 
2 

5 
4 

3 

7 

8 
2 

9 
1 

6 
6 

9 
9 

9 

7 

8 
5 

9 
6 

6 
5 

7 

9 

4 

2 

6 

6 

9 

4 

5 

7 

8 

5 

6 

2 

7 

9 

9 

8 

3 

9 

9 

8 

9 

7 

42 

58 

65 

44 

78 

56 

95 

57 

62 

48 

21 

48 

9 

7 

9 

8 

9 

7 

6 

8 

7 

9 

6 

9 

32 

39 

46 

34 

63 

48 

52 

65 

85 

95 

79 

49 

7 

8 

9 

7 

8 

8 

9 

9 

8 

7 

8 

•8 

7 

4 

7- 

7 

6 

8 

9 

7 

9 

3 

4 

6 

3 

9 

8 

6 

8 

4 

2 

3 

3 

2 

1 

2 

2 

6 

5 

7 

6 

8 

9 

9 

8 

5 

5 

2 

3 

7 

4 

5 

4 

7 

5 

7 

9 

5 

8 

9 

5 

2 

2 

2 

3 

1 

3 

8 

4 

7 

7 

8 

2 

1 

4 

3 

3 

2 

2 

3 

7 

3 

5 

2 

§  39]  ADDITION  17 


3 

8 

6 

6 

8 

4 

4 

8 

9 

7 

2 

4 

9 

9 

7 

4 

8 

6 

8 

4 

7 

9 

5 

7 

7 

3 

7 

8 

4 

6 

3 

6 

4 

4 

6 

9 

27 

45 

72 

86 

45 

39 

48 

67 

54 

59 

75 

93 

6 

8 

9 

3 

6 

7 

5 

5 

4 

8 

2 

5 

6 

2 

1 

6 

4 

3 

5 

6 

6 

2 

8 

7 

65 

84 

93 

82 

49 

29 

56 

87 

76 

86 

45 

52 

9 

7 

6 

5 

3 

4 

7 

8 

7 

2 

9 

6 

3 

9 

5 

7 

2 

3 

2 

3 

4 

4 

1 

4 

NOTE.  Ten  and  twenty  practically  add  themselves  to  any  number ;  hence 
in  adding  columns  of  figures  an  advantage  is  always  secured  by  finding  groups 
aggregating  ten  or  twenty.  To  form  these  groups  it  is  sometimes  advisable  to 
take  up  the  figures  of  a  column  in  irregular  order. 

Thus  in  adding  3,  8,  7,  2,  5,  and  8,  if  3  and  7  are  combined  first,  the  group 
18  is  instantly  seen  ;  then  if  2  and  8  are  next  combined,  the  group  15  and  the 
total  33  is  quickly  obtained. 

In  adding  the  following  numbers,  form  groups  of  ten  and  twenty  wherever 


possible. 

57 

62 

47 

38 

52 

68 

58 

57 

62 

79 

52 

74 

3 

2 

1 

4 

5 

2 

2 

8 

7 

1 

1 

2 

3 

5 

6 

5 

4 

3 

8 

1 

1 

4 

6 

9 

4 

T~ 

TV 

~~ 

~ 

59 

31 

37  ' 

45 

29 

39 

14 

25 

35 

37 

25 

36 

1 

2 

1 

2 

6 

1 

7 

6 

5 

4 

9 

3 

4 

1 

8 

3 

1 

6 

6 

9 

6 

7 

2 

8 

6 

7 

1 

1 

3 

3 

7 

5 

9 

9 

9 

9 

To 

~~ 

~~ 

— 

~s 

- 

4 

1 

4 

3 

2 

7 

6 

9 

4/ 

6 

2 

2 

2 

2 

1 

2 

2 

4 

1 

2 

3 

7 

5 

7 

3 

3 

4 

2 

4 

3 

7 

9 

6 

5 

9 

7 

4 

1 

2 

4 

8 

7 

7 

1 

7 

4 

4 

7 

9 

8 

6 

6 

8 

3 

2 

9 

6 

3 

5 

5 

4 

8 

1 

8 

4 

6 

8 

6 

7 

5 

2 

3 

8 

9 

5 

4 

2 

5 

4 

7 

8 

5 

6 

4 

9 

5 

9 

3 

9 

7 

8 

8 

9 

9 

6 

£V 

18  SIMPLE   NUMBERS  [§§  39-41 

12  11  13  14  15  41  47  56  41  37  59  68 
327433346296 
783  2969  2,  5918 
561849865712 

J>_!_^_^_i    2    2    9   _?   J?   _J?   _f 

40.  Horizontal  Addition.     Numbers,  wfrem  written  in  horizontal 
lines,  as  on  invoices  and  other  business  forms,  may  be  added  with- 
out being  rewritten  in  vertical  columns. 

41.  In  adding  numbers  horizontally,  add  from  left  to  right  and 
then  verify  all  results  by  adding   from  right  to  left.     The  group 
method  may  be  employed  to  advantage  where  numbers  are  written 
horizontally.     The  ability  to  add  horizontally  saves  a  great  deal  of 
time  in  making  out  bills  and  in  performing  other  commercial  opera- 
tions. 

ORAL  EXERCISE 

Add  from  left  to  right  and  review  from  right  to  left  the  following : 
1.   9,  9,  2,  5,  4,  3,  1,  6,  2.  6.   21,  32,  40,  82,  56,  30. 

8.  42,  21,  46,  32,  14,  21.  7.   31,  18,  28,  36,  45,  21. 

3.  52,  46,  35,  72,  68,  50.  8.   67,  61,  60,  63,  62,  65. 

4.  21,  26,  32,  34,  81,  63,  45,  90,  31.      9.   51,  67,  34,  58,  56,  29. 

5.  66,  31,  41, 18,  41,  62,  59,  35,  45.    10.    62,  60,  51,  28,  35,  62. 

11.  How  many  days  in  the  summer  months  ? 

12.  Find  the  sum  of  the  four  numbers  that  may  be  expressed  by 
the  figures  2  and  3 ;  4  and  5 ;  6  and  7. 

13.  Find  the  sum  of  all  the  even  numbers  from  6  to  12  inclusive. 
NOTE.     When  figures  to  be  added  appear  in  consecutive  order,  and  there  is 

an  odd  number  of  them,  the  total  may  be  found  by  multiplying  the  middle 
figure  by  the  number  of  consecutive  figures. 

Thus,  3  +  4  +  5  +  6  +  7  =  5x5  =  25. 

When  any  numbers  appear  in  consecutive  order  their  total  may  be  found 
by  multiplying  one  half  the  sum  of  the  first  and  last  numbers  by  the  number  of 
consecutive  numbers. 

Thus,  14  +  15  +  16  +  17  +  18  =  16  x  6  =  80. 

14-   Find  the  sum  of  all  the  numbers  from  7  to  19  inclusive. 

15.  Find  the  sum  of  all  the  numbers  from  1  to  9  inclusive. 

16.  Find  the  sum  of  all  the  numbers  from  3  to  19  inclusive ;  of 
all  the  numbers  from  5  to  13  inclusive. 


§41] 


ADDITION 


19 


17.  Find  the  sum  of  42,  42,  42,  42,  75. 

NOTE.     When  a  number  is  repeated  several  times  in  any  addition,  the  work 
may  be  shortened  by  multiplication. 

18.  A  man  who  was  born  in  1853  died  when  he  was  forty-nine 
years  old.     In  what"  year  did  he  die  ? 

19.  What  is  the  sum  of  15,  23,  36,  18,  28,  92  ?  of  21,  22,  23,  24, 
25,26,27,28,29,30,31? 

WRITTEN  EXERCISE 

Drill  on  the  following  and  similar  problems  until  correct  results 
can  be  obtained  in  twenty  seconds  or  less. 


1. 

2426^264 
62462148 
64292862 
56259421 
62462962 
52462564 
62469264 
62462942 

2. 
47257386 
52472164 
83492752 
26534721 
23425625 
32462813 
42612542 
78955473 

8. 

27452462 
87950241 
20724065 
86957447 

72757786 
77777777 
88888888 
22222222 

* 

56319217 
48263547 
62519546 
38641948 
95722618 
77554286 
62496246 
62462942 

5. 

72519218 
67482153 
72186349 
39256258 
78295416 
87596357 
21111016 
20407030 

ft 

57264592 
87492165 
48576901 
66875465 
52163441 
10205211 
93758617 
58759218 

Drill  on  the  fol  owing  and  similar  exercises  until  correct  results 
may  be  obtained  in  twenty  seconds  or  less. 

7.        8  9.  10.  11.  12. 

2714  4052  4032  3146  1487  1846 

2652  6021  5061  4219  2116  1092 

1493  1473  4728  2614  4574  1531 

7510  6687  3214  9743  6589  1675 

1126  7214  6010  6478  3752  1832 

4251  9386  5271  2592  1678  1645 

6859  7521  2642  7286  7593  1729 

3114  4268  5537  4924  9164  1011 

7996  7821  6214  6214  7386  4010 

4216  5275  9146  7585  9552  6020 

3114  3942  3910  2137  7829  5190 

6996  4728  1120  7214  3687  1786 

7245  3659  2110  2110  2014  2405 

3865  2854  1640  1016  1730  7216 

5125  7529  2114  4032  3019  4520 

6219  2110  1431  2016  2170  7121 

4346  1011  5214  6147  2590  2514 


20 


SIMPLE   NUMBERS 


T§  41-43 


18.  Show  the  totals  of  the  following  columns  downwards  and 
from  left  to  right.  Prove  the  results  by  adding  the  vertical  and 
horizontal  totals. 


6249 

2145 

2592 

6014 

2172 

4592 

_ 

4625 

1687 

1649 

5019 

1645 

7126 



1872 

1421 

3145 

2041 

1392 

5218 



4124 

3652 

1650 

6215 

1746 

9041 



3635 

1926 

1722 

9013 

7592 

7592 



4216 

4521 

1490 

7016 

6219 

6218 



3417 

1725 

7518 

4110 

5764 

7527 



1641 

1686 

2041 

6211 

2047 

2692 



4356 

4035 

4250 

2140 

6211 

1420 















14.  Complete  the  following  table  by  showing  the  totals  of  the 
columns  vertically  and  horizontally.  Prove  the  work  by  adding  the 
vertical  and  horizontal  totals. 

DEPARTMENTAL  SALES  FOR  THE  WEEK  ENDING  Nov.  15,  1903 


DAYS 

CLOTHING 

DRY  GOODS 

FURNISHINGS 

MILLINERY 

HOUSEHOLD 

UTENSILS 

TOTAL 

Monday 

$790.50 

$988.40 

$126.50 

$256.85 

$496.80 

Tuesday 

640.18 

890.50 

90.18 

420.62 

841.62 

Wednesday 

960.70 

950.40 

75.60 

398.40 

462.60 

Thursday 

490.18 

960.80 

214.90 

425.60 

521.90 

Friday 

930.50 

720.60 

126.70 

396.80 

762.80 

Saturday 

840.15 

989.72 

215.20 

469.65 

925.64 

Total 

42.  The  Two-column  Method  of  Addition.     Some  accountants  are 
very  partial  to  the  two-column  method  of  addition,  claiming  that  it 
is  more  rapid  and  accurate. 

43.  In  adding  two  columns  at  once,  combine  first  the  tens  of  the 
numbers  and  then  the  units. 

Thus,  in  adding  75  and  32  think  of  105  (75  +  30)  and  2,  or  107. 


§  48]  ADDITION  21 

To  illustrate  this  method  of  addition,  take  the  accompanying  example. 
Beginning  with  the  number  46  at  the  top  of  the  column,  add  first  the 
^Q       tens  and  then  the  units  of  the  successive  numbers,  as  follows: 

32  46-f30  =    76;    76  +  2=    78 

65  78  +  60  =  138  ;  138  -f  6  =  143 

51  143  +  60  =  193  ;  193  +  1  =  194 

26  194  +  20  =  214 ;  214  +  6  =  220 

220  In  making  computations  in  this  manner   name   results   only. 

Thus,  beginning  at  the  top  of  the  accompanying  example  and  adding 
downwards,  read  76,  8,  138,  143,  193,  4,  214,  220. 

ORAL  EXERCISE 

1.  Add  the  following  by  double  columns  as  explained  above : 

45  24     52     39     28     57     62     27     33     41     12     13     19     57     92 
39    26     58     56     52     31     34     48     45     37     43     56     38     14     12 

2.  Announce  the  totals  of  the  above  combinations  at  sight  from 
left  to  right  and  from  right  to  left.     Thus,  84,  50,  etc. 

NOTE.  Require  that  this  work  be  done  rapidly.  Drill  on  the  above  and 
similar  combinations  until  the  student  can  announce  the  totals  as  rapidly  as  he 
can  count  1,  2,  3,  etc. 

3.  Add  the  following  by  double  columns,  naming  results  only : 

61  63  82  21  43  19  42  37  51  28  46  27  43  19  24 
39  17  19  23  17  31  24  33  25  52  25  32  27  41  26 
434140366444162872111021111621 

4-  Add  the  numbers  in  problem  3  horizontally  by  the  two-column 
method.  Add  from  left  to  right  and  verify  the  work  by  adding  from 
right  to  left 

5.  Name  results  only  in  determining  the  totals  of  the  following 
by  the  two-column  method : 

28  14  64  48  37  51  45  16  24  81  59  72  27  45  52 

46  26  81  52  43  42  92  41  36  47  31  16  52  92  41 
25  42  95  13  94  18  61  72  33  16  73  41  95  27  92 
92  18  62  24  26  36  43  86  37  52  87  64  65  46  68 
51  32  51  37  51  41  86  14  42  19  49  17  84  85  72 
28  16  28  82  28  35  91  92  48  37  51  86  76  41  14 


22  SIMPLE  NUMBERS  [§§44-47 

44.  Proving  Addition.      The   simplest  way  to  test  the  correct- 
ness of  addition  is  to  add  the  columns  a  second  time  in  reverse 
order. 

45.  Accountants  who  have  to  add  very  long  columns  of  figures 
frequently  begin  at  the  right-hand  column  and  write  on  a  piece  of 
waste  paper  the  full  sum  of  each  column  added,  and  then,  to  verify 
the  work,  begin  at  the  left  side  and  add  the  columns  in  reverse 
order,  again  writing  the  full  sums  on  a  piece  of  waste  paper.     If  the 
sum  of  the  totals  shown  by  the  first  addition  is  the  same  as  the  sum 
of  the  totals  shown  by  the  second  addition,  the  work  is  assumed  to 
be  correct. 

46.  The  accountant  not  infrequently  has  to  perform  his  work 
amid  more  or  less  confusion.     In  adding  long  columns,  if  this  method 
is  employed,  he  can  be  interrupted  or  can  leave  his  work  for  a  time 
and  resume  it  again  without  examining  in  detail  the  columns  which 
have  already  been  completed. 

47.  This  method  of  proving  addition  is  illustrated  in  the  follow- 
ing example  and  solution  : 

SOLUTION.     Beginning  with  the  right-hand  column  and  adding  downwards, 
the  total  is  28.     Write  28  to  the  right  of  the  numbers  added,  or  on  a  piece  of 

waste  paper  ;  without  carrying  add  the  next 

19  4225  28       column,  and  the  total  is  15,  which  should  be 

17  6248  15          written  as  shown  in  the  accompanying  illus- 

17  tration  ;  add  the  next  column  without  carry- 


1  Q  »  an(*  tlie  tota^  *s  ^  '  ad(*  tne  next  c°lumn» 

and  the  total  is  19.     The  sum  of  these  totals 


20878      2130      20878      is  20878. 

20878  VERIFICATION.     Beginning  with  the  left- 

hand  column  and  adding  upwards,  the  total 

is  19.  Write  this  to  the  left  of  the  figures  to  be  added  or  on  a  piece  of  waste 
paper,  as  shown  in  the  accompanying  illustration  ;  without  carrying  add  the  next 
column,  and  the  total  is  17,  which  write  as  shown  in  the  illustration  ;  add  the 
next  column,  and  the  total  is  15  ;  the  next,  and  the  total  is  28.  The  sum  of 
these  totals  is  20878,  or  the  same  as  found  by  the  first  addition  ;  hence  it  is 
assumed  that  the  work  is  correct. 

WRITTEN  REVIEW 

1.  In  the  following  statement  add  the  columns  downwards  and 
from  left  to  right,  and  then  prove  the  work  by  adding  the  vertical 
and  horizontal  totals. 


§47] 


ADDITION 


23 


STATE   ASSESSMENTS 


YEAR 

ARMORIES 

METROPOLI- 
TAN 

ABOLITION 
OP  GRADE 

METROPOLITAN 

HIGHWAYS 

TOTAL 

SEWER 

CROSSINGS 

WATER 

1895-1896 

$21,498.29 

$59,702.19 

$25,811.94 

$285,600.54 

$161.67 

1890-1897 

28,056.27 

119,321.10 

45,583.53 

211,901.92 

100.45 

1897-1898 

28,056.27 

95,421.14 

62,677.92 

199,900.41 

571.94 

1898-1899 

34,223.15 

75,753.81 

56,854.31 

258,990.00 

153.23 

1899-1900 

34,223.15 

129,773.27 

71,662.03 

411,861.54 

101.82 

1900-1901 

34,223.15 

12,625.73 

131,074.00 

578,696.96 

68.78 

Total 

2.   Complete  the  following  sales  sheet, 
ing  the  vertical  and  horizontal  totals. 


Prove  the  work  by  add- 


SUMMARY  OF  DAILY  SALES 


JULY  2 

SHOES 

GLOVES 

HATS 

DRESS 
GOODS 

CLOTHING 

TOTAL 

A  to  D  Ledger 

$237.31 

$126.92 

$132.16 

$263.64 

$423.09 

E  to  H  Ledger 

228.80 

140.75 

110.25 

357.18 

387.75 

I  to  L  Ledger 

238.84 

231.78 

106.35 

676.83 

627.71 

M  to  P  Ledger 

143.54 

157.57 

161.69 

382.55 

641.23 

Q  to  T  Ledger 

848.49 

657.02 

510.45 

510.59 

651.45 

U  to  Z  Ledger 

556.51 

213.19 

388.54 

811.82 

680.29 

Total 

3.  The  sales  of  a  dry  goods  house  for  the  week  ending  Nov.  22, 
1903,  were  as  follows:  Monday,  domestics,  $540.10;  notions, 
$325.85;  woolens,  $864.98;  dress  goods,  $325.78.  Tuesday,  do- 
mestics, $995.85;  notions,  $419.62;  woolens,  $919.10;  dress 
goods,  $146.84.  Wednesday,  domestics,  $975.89;  notions,  $853.64; 
woolens,  $  1659.89 ;  dress  goods,  $  1259.89.  Thursday,  domestics, 
$856.74;  notions,  $459.13;  woolens,  $756.85;  dress  goods,  $588.74. 
Friday,  domestics,  $862.47;  notions,  $817.39;  woolens,  $1249.86; 
dress  goods,  $1560.84.  Saturday,  domestics,  $1529.84;  notions, 
$915.62;  woolens,  $958.22;  dress  goods,  $1079.54. 

Arrange  these  facts  in  tabular  form,  in  six,  columns,  with  proper 
headings.  Show  (a)  the  total  sales  for  each  department,  (b)  the 
total  daily  sales,  and  (c)  the  total  sales  for  the  week. 


SIMPLE   NUMBERS 


[§  47 


4.  The  records  of  a  city  post  office  show  the  following  mail  for 
one  week  :  Monday,  registered  letters,  725 ;  ordinary  letters,  15,279 ; 
postal  cards,  2147;  book  packets,  963;  parcels,  184;  newspapers, 
26,419.  Tuesday,  registered  letters,  461 ;  ordinary  letters,  12,365 ; 
postal  cards,  2011;  book  packets,  395;  parcels,  416;  newspapers, 
21,936.  Wednesday,  registered  letters,  369 ;  ordinary  letters,  16,285 ; 
postal  cards,  1989;  book  packets,  618;  parcels,  365;  newspapers, 
23,162.  Thursday,  registered  letters,  8490 ;  ordinary  letters,  14,317  ; 
postal  cards,  416;  book  packets,  562;  parcels,  213;  newspapers, 
23,164.  Friday,  registered  letters,  959;  ordinary  letters,  25,162; 
postal  cards,  2116;  book  packets,"  475 ;  parcels,  163;  newspapers, 
22,790.  Saturday,  registered  letters,  416;  ordinary  letters,  11,259; 
postal  cards,  659;  book  packets,  384;  parcels,  175;  newspapers, 
21,218. 

.Arrange  these  facts  in  tabular  form,  in  eight  columns,  with  proper 
headings.  Find  (a)  the  total  number  of  separate  pieces  of  mail  for 
each  day,  (6)  the  total  number  of  pieces  of  each  class,  and  (c)  the 
total  number  of  pieces  for  the  week. 


Copy  or  write 

from  dictation 

and  find  the  sums 

of  the  following: 

5.      .*2x* 
92451826 
40159061 

6. 
24164290 
72154031 

7. 
32169528 
62169528 

8. 
12345678 
28968457 

52192165' 

16941762 

62195437 

10475631 

87965421 

15304693 

65954370 

20047509 

74926587  \ 
59346599  ^ 

31462845 
32168492 

65109011 
52416011 

33715586 
88475*621 

92657788 
65945876 

11141017 
21411731 

10401721 
41627428 

78991047 
74839101 

92517496 

17283142 

31426357 

10108765 

93479491 

65493762 

21407110 

56461086 

59627488 

58911476 

11169042 

77562345 

95178654 

72491368 

25172825 

87693421 

72958649 

72159072 

41627598 

24683157 

58721985 

21311510 

66901080 

36912141 

58759271 

21411631 

72164010 

11354678 

21864925 

47293742 

69957788 

10019087 

17264592 

40171650 

28521654 

98798778 

18259015., 

21101670 

29364124 

76453111 

§§  48-54]  SUBTRACTION  25 

SUBTRACTION 

48.  Subtraction  is  the  process  of  finding  the  difference  between 

two  numbers. 

49.  The  subtrahend  is  the  number  to  be  subtracted. 

50.  The  minuend  is  the  number  from  which  the  subtrahend  is  to 
be  subtracted. 

51.  The  remainder  or  difference  is  the  number  obtained  by  sub- 
traction. 

52.  The  sign  —  signifies  subtraction  and  is  read  minus  or  less. 

When  the  sign  of  subtraction  is  placed  between  two  numbers,  it  indicates  that 
the  number  written  after  it  is  to  be  taken  from  the  one  written  before  it 

53.  Numbers  written  within  a  parenthesis  (  ),  under  a  vinculum 
,  or  separated  by  the  sign  of  multiplication  x ,  are  to  be  considered 

together. 

Thus,  16  -  (4  +  2)  or  16  -  4  +  2  signifies  that  the  sum  of  4  and  2  is  to  be 
subtracted  from  16 ;  16  -  4  x  2  signifies  that  the  product  of  4  and  2  is  to 
be  subtracted  from  16. 

54.  General  Principles.     1.   Only  the  same  orders  of  units  of  like 
numbers  can  be  subtracted. 

2.  The  sum  of  the  subtrahend  and  remainder  is  equal  to  the 
minuend. 

ORAL  EXERCISE 

1.  Subtract  by  4's  from  44  to  0 ;  from  39  to  3. 

2.  Subtract  by  6's  from  49  to  1 ;  from  78  to  0. 

3.  Subtract  by  5's  from  135  to  0 ;  from  121  to  1. 

4.  Subtract  by  7's  from  38  to  3 ;  from  64  to  1 ;  from  44  to  2. 

5.  Subtract  by  8's  from  91  to  3 ;  from  55  to  7 ;  from  37  to  5. 

6.  Subtract  by  9's  from  131  to  23;  from  57  to  12;  from  95  to  5. 

7.  Subtract  by  15's  from  90  to  0;  from  120  to  0;   from  76  to  1. 

8.  Subtract  by  13's  from  41  to  2;  from  57  to  5;  from  63  to  11. 

9.  Subtract  by  ll's  from  88  to  0;  from  72  to  6;  from  91  to  3. 
10  Subtract  by  12's  from  54  to  6;  from  128  to  8;  from  145  to  t 


26  SIMPLE  NUMBERS  [§55 

55.   Example.     Find  the  difference  between  348  and  185. 

348  SOLUTION.     Since  only  units  of  the  same  order  can  be  subtracted, 

185      write  units  under  units,  tens  under  tens,  and  hundreds  under  hun- 

Jg3      dreds,  and  draw  a  line  beneath.     Beginning  at  the  right,  5  units  from 

8  units  leaves  3  units.     Write  3  under  the  column  of  units.     Since 

8  tens  cannot  be  taken  from  4  tens,  transform   1   of  the  3  hundreds  into 

10  tens,  and  add  it  to  the  4  tens,  making  14  tens ;  then,  8  tens  from  14  tens 

leaves  6  tens,  which  write  under  the  column  of  tens.    Since  1  of  the  3  hundreds 

has  been  taken,  there  are  only  2  hundreds  remaining  ;  then,  1  hundred  from 

2  hundreds  leaves  1  hundred.     The  difference  between  the  two  numbers  given 

is,  therefore,  163. 

In  practice  think  only  of  results  and  write  them  without  hesitating.    Thus, 
in  the  above  problem  write  or  think  only  3,  6,  1. 


ORAL  EXERCISE 

1.  From  what  number  must  we  subtract  $2.54  to  have  $7,46 
remaining  ? 

2.  If  I  pay  $375  for  a  carriage  and  sell  it  at  a  loss  of  $73.75, 
how  much  do  I  receive  for  it  ? 

S.  The  smaller  of  two  numbers  is  96;  their  difference  is  46. 
What  is  the  larger  number  ? 

4-  Pronounce  at  sight  the  difference  between  the  numbers  in 
each  of  the  following  groups. 

79          62          56          85          67          78        89          98          67 
34          27          47          19          21          49       fc!3          14          12 

105        107        108        106        127        168        99        119        121 

97          53          38          56          77          48        38          47          69 

5.  Pronounce  at  sight  the  difference  between  the  numbers  in 
each  of  the  following  groups. 

The  subtrahend  is  placed  above  the  minuend  in  order  to  give  practice  in  find- 
ing the  difference  between  numbers  that  are  so  arranged.  If  one  is  not  able  to 
subtract  in  this  manner,  he  is  frequently  required  to  rearrange  the  numbers  on 
separate  paper  to  subtract  them.  This  is  a  waste  of  time,  since  by  a  little 
practice  one  can  readily  subtract  numbers  that  are  not  regularly  arranged. 


§§  55-60]  SUBTRACTION  27 

79     58  72  29    47     57  62  49  26 

92    164  149  132    98    124  126  93  78 

18    123  93  47    244    169  158  137  214 

54    246  186  94    488    239  248  267  264 


SHORT  METHODS 

56.  The  complement  of  a  number  is  the  difference  between  such 
number  and  a  unit  of  the  next  higher  order. 

Thus,  the  complement  of  6  is  4,  since  4  is  the  difference  between  6  and  10,  or 
1  ten,  a  unit  of  the  next  higher  order  than  6  ;  the  complement  of  83  is  17,  since 
17  is  the  difference  between  83  and  100,  or  1  hundred,  a  unit  of  the  next  higher 
order  than  83. 

57.  Two  numbers  whose  sum  is  equal  to  a  unit  of  the  next  higher 
order  are  called  complementary  numbers. 

Thus,  209  and  791  are  complementary  numbers,  since  their  sum  is  equal  to 
1000  ;  2467  and  7633  are  complementary  numbers  since  their  sum  is  equal  to 
10000. 

58.  If  two  numbers  of  more  than  one  figure  are  complementary 
numbers,  the  sum  of  their  units  figures  is  10,  and  of  each  of  their 
corresponding  higher  orders,  9. 

Thus,  642  and  358  are  complementary  numbers  ;  the  sum  of  the  units  figures 
is  10,  and  the  sum  of  the  figures  in  the  corresponding  higher  orders  is  9. 

59.  The  foregoing  principle  may  be  applied  to  advantage  in  mak- 
ing change.     Since  we  read  numbers  from  left  to  right,  it  is  gener- 
ally best  in  making  change  to  begin  at  the  left  to  subtract.     In 
beginning  at  the  left  to  subtract,  take  1  from  the  number  of  units  of 
the  highest  order  in  the  minuend,  and  regard  each  of  the  lower  orders 
as  9  except  the  last,  which  must  be  regarded  as  10. 

60.  Example.     A  gave  a  twenty-dollar  bill  in  payment  for  an 
account  of  $14.72.     How  much  change  should  he  receive? 

SOLUTION.  Begin  at  the  left  to  subtract.  1  from  the  highest 
order  in  the  minuend  leaves  1.  1  from  1  leaves  0.  4  from  9 
leaves  5,  which  write  in  the  units'  column.  7  from  9  leaves  2, 
which  write  in  the  tenths'  column.  2  from  10  leaves  8,  which 
write  in  the  hundredths'  column.  The  result  is  $5.28. 


2#  SIMPLE   NUMBERS  [§§  00-62 

DRILL  EXERCISE 
By  inspection,  find  the  difference  between  the  following  numbers : 

400      600      300        700        900      1000      100      200      300      500 
132      175        86        263        458        532        52        31        57      138 


$1.00      $2.00     $3.00      $4.00      $5.00      $8.00      $7.00      $8.00 
.39         1.15         1.17         1.58        2.21         2.39         5.36         1.37 


$20.00      $20.00      $10.00      $30.00      $30.00      $40.00      $50.00 
2.59          8.76  5.72         28.61         29.57         25.86  6.78 

Subtract  each  of  the  following  numbers  from  $2.00:  27^,  52^, 
89£  $1.52,  $1.13,  $1.41,  $1.59,  85^  41  £  37  £  56^,  18£  97  £ 

If  $100  is  offered  in  payment  for  each  of  the  following  accounts, 
what  amount  of  change  should  be  returned  ?  $  25.95,  $  85.67,  $  37.54, 
$92.18,  $65.51,  $87.75,  $69.52,  $18.75,  $37.58,  $88.13,  $71.15, 
$41.30,  $39.18,  $25.72. 

Exercises  similar  to  the  above  should  be  continued  until  correct  results  can  be 
given  without  a  moment's  hesitation. 

61.  Frequently  an  accountant  finds  it  desirable  to  take  the  sum  of 
several  numbers  from  the  sum  of  several  other  numbers  without 
transferring  the  totals  from  the  books  of  record  to  separate  paper. 
The  following  explanations  will  be  found  suggestive  of  short  cuts 
which  may  be  employed  to  advantage  in  such  cases. 

62.  Examples.     1.   From  24,794  subtract  the  sum  of  4159,  6490, 
and  4462. 

24794  SOLUTION.     For  convenience  write  the  numbers  under  each 

, ..  -Q      other  with  the  minuend  set  off  from  the  subtrahend  by  a  straight 
line.     Beginning  at  the  right  and  adding  the  units  of  the  subtra- 
hend the  sum  is  11,  which,  subtracted  from  14  (the  next  higher 
4462      number  ending  with  4),  leaves  3,  the  units  of  the  required  result. 
9683  The  sum  of  the  figures  in  the  tens'  column  plus  1  (the  number 

of  tens  added  to  the  minuend  in  the  previous  subtraction)  is  21 , 
which,  subtracted  from  29  (the  next  higher  number  ending  with  9),  leaves  J,  the 
tens'  figure  of  the  required  result. 


§  62]  SUBTRACTION  29 

The  sum  of  the  figures  in  the  hundreds'  column  plus  2  (the  number  of  hun- 
dreds added  to  the  minuend  in  the  previous  subtraction)  is  11,  which,  subtracted 
from  17,  leaves  6,  the  hundreds'  figure  of  the  required  result. 

The  sum  of  the  figures  in  the  thousands1  column  plus  1  (the  number  of 
thousands  added  to  the  minuend  in  the  previous  subtraction)  is  15,  which, 
subtracted  from  24,  leaves  9,  or  the  thousands*  figure  of  the  required  result. 

2.  The  gross  weights  and  tares  of  5  barrels  of  sugar  are  as  fol- 
lows :  319-19,  322-21,  311-17,  322-19,  329-21  pounds.  Find  the 
net  weight. 

SOLUTION.  In  billing,  the  above  numbers  would  be  written  horizontally  as 
follows:  319-19,  322-21,  311-17,  322-19,  329-21. 

The  minuend  is  the  gross  weight,  and  the  subtrahend  is  the  tare.  Adding 
the  units  of  the  subtrahend  horizontally  the  sum  is  27,  which,  subtracted  from 
the  next  higher  order  of  units  (30),  leaves  3.  3  added  to  the  units  of  the  minu- 
end equals  26.  Write  6  as  the  units  of  the  net  weight.  Since  the  tens  of  the 
subtrahend  are  1  more  than  the  tens  of  the  minuend,  add  1  to  the  next  higher 
order  of  the  subtrahend,  or  subtract  1  from  the  next  higher  order  of  the  minu- 
end. Add  1  to  the  tens  of  the  subtrahend,  and  the  result  is  8,  which,  subtracted 
from  the  next  higher  order  of  units  (10),  leaves  2.  Adding  2  to  the  tens  of  the 
minuend,  the  result  is  10.  Write  0  as  the  tens  of  the  net  weight.  Since  the  tens 
of  the  minuend  are  the  same  as  the  tens  of  the  subtrahend,  there  is  nothing  to 
carry.  Adding  the  hundreds  of  the  minuend,  the  result  is  15.  Write  15  as  the 
hundreds  of  the  net  weight.  The  net  weight  is  then  1506  pounds. 

NOTE.  In  billing  where  items  are  listed  as  gross  weight  and  tare  the  above 
process  will  be  found  especially  helpful.  Sufficient  practice  should  be  required 
to  give  the  student  facility  in  making  the  extensions  properly. 

This  principle  may  also  be  used  to  advantage  hi  finding  the  balances  of 
ledger  accounts. 

WRITTEN  EXERCISE 

NOTE.  In  the  first  four  problems  below  the  gross  weight  in  pounds  is  written 
to  the  left  of  the  hyphen  and  the  tare  in  pounds  to  the  right  of  the  hyphen. 
Find  the  net  weight  as  explained  in  Example  2,  above. 

1.  10  casks  of  hams,  392-67,  412-71,  402-71,  411-67,  408-68, 
425-71,  400-69,  399-70,  398-71,  426-68. 

&  6  baskets  pork  loins,  312-49,  301-56,  297-48,  415-43, 312-49, 
314-56. 

8.  4  tubs  of  lard,  71-14,  70-15,  69-14,  62-15. 

4.  8  casks  shoulders,  428-19,  322-21,  327-19,  311-17,  314-17, 
315-18,  317-21,  342-24. 


SIMPLE   NUMBERS 


[§62 


By  either  of  the  short  methods  explained  in  62  find  the  balances 
of  the  following  accounts. 

6. 

*Dr.  The  Union  Bank,  in  account 


6. 
*Dr.  The  Union  Bank,  in  account  with 


7. 


£1. 

_A*  -. 

/  /^  ^ 

tf  0 

"faL,, 

260 

/  2. 

'         ^         » 

^2-^X 

^ 

3  / 

&4JL 

&r 

2-0 

J^ 

72/7 

^  / 

tMA 

n 

^  / 

/!*/ 

£A 

?/ 

00 

7? 


7? 


720  KS 


§§  62-64] 


SUBTRACTION 
8. 


81 


,-fc 


What  is  the  balance  in 

Deposit  Aug.  17,  $65.98. 
What  is 


Find  the  balances  of  the  following  bank  accounts  without  using 
pen  or  pencil  except  to  write  the  results. 

9.   Balance  in  bank  June  1,  $  650.40.     Checks  from  June  1  to 
July  1,  $  145.20,  $  14.90,  $  60.50,  $  20.40. 
\>auk  July  1  ? 

.70.   Balance  in  bank  Aug.  15,  $  695.40. 
Checks  from  Aug.  15  to  Sept.  1,  $  146.20,  $  90.50,  $60.95. 
the  balance  in  bank  Sept.  1  ? 

11.  Balance  in  bank  Jan.  15,  $460.40.  Deposit  Jan.  20, 
$  152.65.  Checks  from  Jan.  15  to  Feb.  1,  $172.40,  $  14.90,  $  16.95, 
$  40.65.  What  is  the  balance  in  bank  Feb.  1  ? 

63.  Combining  Addition  and  Subtraction  in  One  Process.     When  a 
number,  or  the  total  of  several  numbers,  is  to  be  taken  away  from 
the  total   of   several   other  numbers,   the   two   processes   may   be 
combined  as  in  62,  or  as  shown  in  the  following  examples. 

64.  Examples.     1.   From  the  sum  of  12  and  6  take  3. 

SOLUTION.  Adding  12  to  6  we  have  18.  It  is  self-evident  that  18  —  8  is 
equivalent  to  18+  (10  -  3)-  10. 

When  a  number  is  both  added  to  and  subtracted  from  any  quantity, 
the  value  of  the  quantity  is  not  changed. 

Applying  this  principle  in  solving  the  above  problem,  mentally 
take  3  from  10,  add  the  difference  to  6  aid  12,  and  subtract  10  from 
the  result.  Thus,  7,  25,  15,  the  required  result. 


32  SIMPLE   NUMBERS 


2.    From  the  sum  of  827  and  534  subtract  356. 


[§§  C4-67 


SOLUTION.  Arranging  the  numbers  horizontally  and  adding  the  units,  by 
naming  the  results  only,  we  have  from  the  right  4  (10^6),  8,  15,  5  (15—  10) 
to  write  as  the  first  figure  of  the  required  result.  Adding  the  tens,  we  have  5 
(10  —  5),  8,  10,  0  (10—  10)  to  write  as  the  tens  of  the  required  result.  Adding 
the  hundreds,  we  have  7  (10  -  3),  12,  20,  10  (20  -  10)  to  write  as  the  hundreds 
of  the  required  result.  The  final  result  is,  therefore,  1005. 

3.  From  the  sum  of  729  and  642  subtract  211. 

SOLUTION.  9,  11, 20,  or  0  to  write  and  1  to  carry  to  the  minuend.  10  (9  -f  1 
carried),  14,  16,  or  6  to  write.  8,  14,  21,  or  11  to  write.  The  final  result  is, 
therefore,  1160. 

4.  From  the  sum  of  321  and  811  subtract  369. 

SOLUTION.  1,  2,  3,  or  3  to  write  and  1  to  subtract  from  the  tens  of  the  minu- 
end. 3  (4  —  1),  4, 6,  or  6  to  write  and  1  to  subtract  from  the  minuend.  6  (7  —  1), 
14,  17,  or  7  to  write.  The  result  is,  therefore,  763. 

NOTE.  In  this  class  of  work  grouping  may  be  used  to  advantage.  To  show 
every  step  in  the  process  the  results  in  the  above  solutions  were  determined  with- 
out grouping. 

65.  Hence  the  following  rule : 

Take  each  order  of  units  in  the  subtrahend  from  10,  add 
the  difference  to  the  same  order  of  units  in  the  minuend, 
and  deduct  10  from  the  result  obtained. 

In  adding  any  order  of  units  if  the  result  is  less  than  20,there  is  nothing  to 
carry  to  the  next  higher  order  in  the  minuend  ;  if  the  sum  is  20  or  more,  there  is 
always  something  to  carry  to  the  next  higher  order  in  the  minuend  ;  if  the  sum 
is  less  than  10,  there  is  1  to  subtract  from  the  next  higher  order  in  the  minuend. 

Thus,  if  the  sum  of  any  order  of  units  is  a  number  from  10  to  19  inclusive, 
carry  nothing  ;  a  number  from  20  to  29  inclusive,  carry  1 ;  a  number  from  1  to  9 
inclusive,  subtract  1. 

• 

66.  Individual  Ledger  Balances.    The  above  method  is  particularly 
helpful  in  making  extensions  on  a  banking  individual  ledger. 

67.  Example.    Find  the  balance  to  the  credit  of  D.  Roe  in  the 
following  bank  account : 


NAMES 

BALANCES 

CHECKS 

DEPOSITS 

BALANCES 

D.  Roe 

692 

85 

146 

25 

625 

42 

§67] 


SUBTRACTION 


33 


SOLUTION.  The  first  column  shows  the  balance  on  deposit,  the  second  the 
amount  withdrawn  by  checks,  and  the  third  the  amount  deposited.  The  sum  of 
the  old  balance  and  the  deposits  is  therefore  the  minuend,  and  the  sum  of  the 
checks  the  subtrahend.  Employing  the  principles  just  explained,  the  balance  of 
the  foregoing  account  may  be  determined  mentally,  as  follows  : 

2,    7  (2  4- 10  -  5),  12.     Write  2  in  the  new  balance  column. 

4,  12  (4  -j- 10  -  2),  20.     Write  0  in  the  new  balance  column  and  carry  1. 

6  (5  -f  1  carried),  10  (6  -f  10  —  6),  12.    Write  2  in  the  new  balance  column. 

2,   8  (2  -f- 10  -  4),  17.     Write  7  in  the  new  balance  column. 

6, 15  (6  -f- 10  - 1),  21.     Write  11  in  the  new  balance  column. 

The  new  balance  is,  therefore,  $1172.02. 


WRITTEN  EXERCISE 

1.  Copy  or  write  from  dictation  the  following  individual  ledger 
accounts  and  find  the  new  balances  without  using  pen  or  pencil 
•except  to  write  the  results.  After  extending  the  new  balances  add 
the  old  balances,  checks,  deposits,  and  new  balances,  respectively. 
Prove  the  work.  The  sum  of  the  total  old  balances  and  the  total 
deposits  minus  the  total  checks  .should  equal  the  total  new  balances. 


NAMES 

BALANCES 

CHECKS 

DEPOSITS 

BALANCES 

Allen,  E.  W. 

962 

59 

421 

65 

875 

90 





Briggs,  C.  W. 

725 

42 

126, 

42 

215. 

95 



— 

Comer,  L.  M. 

826 

54 

217 

47 

421 

66. 



— 

Day,  O.  D. 

924 

54 

413 

86 

966 

75 



— 

Emery,  A.  L. 

592 

87 

436 

58 

297 

62 



— 

Foley,  B.  E. 

726 

88 

315 

92 

496 

87 



— 

Good,  J.  I. 

925 

43 

413 

86 

575 

94 



— 

Hall,  L.  0. 

1426 

88 

613 

92 

726 

48, 



— 

Irwin,  Chas.  E. 

1217 

95 

214 

86 

926 

45 



— 

Jones,  Chas.  H. 

725 

77 

216 

54 

818 

72 



— 



— 

' 

— 



— 



— 

t .  In  the  following  account  find  (a)  the  total  checks,  and  (6)  the 
new  balances.     Prove  the  work. 

- 


- 


7 


34 


SIMPLE   NUMBERS 


[§67 


NAMES 

BALANCES 

CHECKS  IN 
DETAIL 

TOTAL 

CHECKS 

DEPOSITS 

BALANCES 

Ames,  D.  T. 

9241 

10 

126 

95 



— 

1400 

00 



— 

Ballou,  M.  T. 

6418 

40 

200 
216 

00 
10 



— 

700 
900 

00 
00 



— 

Collins,  W.  T. 

1421 

19 

500 
417 

00 
40 



— 

920 
1240 

00 
10 

•  

— 

Dorman  &  Co. 

2146 

11 

200 
711 

00 
40 



— 

1750 

92 



— 

Evans  &  Son 

1492 

20 

400 
240 

00 
10 



— 

1120 

00 



— 

Farley  Bros. 

1742 

20 

410 
920 

00 
19 



— 

1750 

00 

Grant  &  Snow  Co. 

2114 

90 

750 

00 

.  

— 

8710 
2500 

00 
00 



— 

Hall  &  Smith 

6218 

10 

1200 

00 





1100 

09 

,  



1460 

41 

Irwin,  J.  T. 

1721 

10 

200 
1140 

00 

80 



— 

1400 

62 



— 

Jamison,  M.  I. 

4216 

91 

600 

00 



— 

1721 

42 



— 



— 

. 

— 



— 



— 



ORAL  REVIEW 

1.  From  100  take   15;  25;  42;  16;  73;  81;  19;  16;  14;  22; 
33;  45;  55;  65;  72;  87;  64;  47;  35;  51;  17. 

2.  I  gave  a  fifty  dollar  bill  in  payment  for  an  account  of  $  23.45. 
How  much  change  should  I  receive  ? 

NOTE.  In  making  change  it  is  always  advisable  to  determine  the  amount 
by  subtraction  and  then  to  verify  the  result  by  addition.  Thus,  if  $  10  is  received 
in  payment  for  a  bill  of  $7.42,  by  inspection,  determine  the  balance  and  prove 
the  result  by  adding  to  the  amount  purchased  the  change  counted  out.  If  the 
amount  of  the  payment  is  obtained  by  such  addition,  the  result  is  assumed  to 
be  correct. 


7.  $30- $22.79  =  ? 

8.  $15- $11.68=? 

9.  $25- $23.75=  ? 


10.   $5-$1.03  +  $2.17 


S.  $10- $4.17=? 

4.  $15-  $2.77=? 

5.  $20-$1.75'+$2.16=? 

6.  $20- $2.95=? 

11.  From  $3  take  89^;  74#;  67^;  83^;  74^;  18^;  24^; 
';  68^;  38^;  27^;  52^.  ' 

12.  In  an  account  with  Charles  Spencer  a  payment  of  $  17  which 
he  makes  is  erroneously  charged,  instead  of  credited,  to  his  account. 
What  is  the  error  in  the  balance  of  his  account  ?    Explain. 


67] 


SUBTRACTION 


35 


13.   By  inspection,  find  the  difference  between  the  following : 
900  400  2000  800  5000  3000 

9  52  172  24  57  127 


$100.00 
14.65 

$200.00 
15.65 

$400.00 
17.24 

$50.00 
5.21 

$100.00 
11.45 

$200.00 
15.65 

$500.00 
65.95 

$300.00 
11.42 

$600.00 
18.52 

$700.00 
17.25 

$  1000.00 
127.50 

$200.00 
18.50 

$150.00 
1.92 

$100.00 
19.00 

$7000.00 
18.59 

$100.00 
7.92 

$3000.00 
15.49 

$10.00 
7.92 

WRITTEN  REVIEW 

1.  A  has  $950,  which  is  $275  more  than  I  have,  and  I  have 
$  300  more  than  B.     How  much  have  we  together  ? 

2.  A  and  B  together  owe  me  $9275;  A  owes  me  $3150.     After 
paying  me  $  1900  on  account,  how  much  does  B  still  owe  me  ? 

3.  A  produce  dealer  bought  200  barrels  of  apples  for   $415. 
Had  he  received  $  75  more  in  selling  them  his  gain  would  have  been 
equal  to  the  amount  originally  paid  for  the  apples.     What  amount 
was  received  from  the  sale  of  the  apples  ? 

4-  A  furniture  dealer  bought  a  stock  of  goods  amounting  to 
$5216.  After  selling  goods  amounting  to  $4917,  he  took  an  ac- 
count of  stock  and  found  that  he  had  furniture  on  hand  amount- 
ing to  $  1937.  Did  he  gain  or  lose,  and  how  much  ? 

5.  A  retail  hardware  dealer  bought  merchandise  amounting  to 
$  1249.     After  selling  from  this  stock  articles  amounting  to  $  842, 
he  took  an  account  of  the  stock  remaining  unsold  and  found  that  it 
was  worth  $  311.     Did  he  gain  or  lose,  and  how  much  ? 

6.  At  the  close  of  the  business,  July  1,  a  merchant  had  cash 
in  the  safe  amounting  to  $314.     July  2  he  received  from  sales  $  526 ; 
on  account,  $435 ;  the  cash  in  the  safe  at  the  close  of  July  2  amounted 
to  $  219.     What  ^yere  the  total  disbursements  for  July  2  ? 

7.  A  father  divided  his  farm,  consisting  of  675  acres,  among  his 
three  sons,  Harvey,  William,  and  Albert.     Harvey  received  75  more 
acres  than  William,  who  received  225  acres,  and  to  Albert  was  given 
the  remainder.     How  many  acres  were  given  to  Albert  ? 


36  SIMPLE  NUMBERS  f§§  68-76 

MULTIPLICATION 

68.  Multiplication  is  the  process  of  taking  one  of  two  numbers  as 
many  times  as  there  are  units  in  the  other. 

69.  The  multiplicand  is  the  number  that  is  to  be  taken  a  required 
number  of  times. 

70.  The  multiplier  is  the  number  which  indicates  how  many 
times  the  multiplicand  is  to  be  taken  or  multiplied. 

71.  The  product  is  the  number  obtained  by  multiplication. 

72.  The   sign  x  signifies   multiplication,  arid  is  read   times  or 
multiplied  by. 

The  sign  x  is  read  times  when  the  multiplier  precedes  the  multiplicand  and 
multiplied  by  when  the  multiplicand  precedes  the  multiplier. 

73.  General  Principles.     1.  The  multiplier  always  signifies  a  num- 
ber of  times,  and  is  an  abstract  quantity. 

2.  The  multiplicand  may  be  either  an  abstract  or  a  concrete 
number. 

3.  The  product  always  has  the  same  name  as  the  multiplicand. 

74.  Factors  are  the  numbers  used  in  obtaining  a  product. 

75.  The  numerical  result  of  one  number  by  another  is  the  same 
whichever  factor  is  regarded  as  the  multiplier.     The  above  general 
principles  are  to  be  recognized  only  in  explanations  of  work  done. 

For  illustration,  take  the  following  example  : 

If  one  barrel  of  apples  cost  $3,  what  will  125  barrels  cost? 

SOLUTION:  Since  1  barrel  of  apples  cost  $3,  125  barrels  will  cost  125  times 
$  3,  which  is  $  375. 

We  cannot  multiply  125  by  $3,  but  since  3  times  125  times  is  equal  to  125 
times  3,  we  may  interchange  the  factors  and  have  3  tim°«  $  125.  The  product, 
it  will  be  observed,  is  the  same  in  either  case.  Hence, 

An  interchange  of  the  factors  in  any  multiplication  does  not  affect 
the  product. 

76.  In  multiplying  one  number  by  another,  always  use  the 
smaller   quantity   as   the   multiplier.      It   should   be   remembered, 
however,  that  the  product  always  has  the  same  name  as  the  true 
multiplicand. 


§§  70-78] 


MULTIPLICATION 


37 


MULTIPLICATION  TABLE 


1 

a 

.'{ 

4 

5 

0 

7 

8 

0 

10 

11 

12 

13 

14 

15 

10 

17 

IS 

111 

20 

21 

22 

2:J 

24 

25 

2 

4 

6 

8 

10 

1-2 

14 

16 

18 

20 

22 

21 

26 

28 

30 

82 

84 

86 

38 

40 

42 

-14 

46 

48 

50 

8 

6 

9 

12 

ir. 

18 

21 

•24 

27 

8Q 

88 

86 

89 

42 

46 

48 

51 

51 

5T 

M 

68 

66 

69 

72 

75 

4 

8 

12 

16 

so 

-24 

28 

32 

36 

40 

41 

•IS 

62 

56 

(id 

64 

68 

72 

T6 

Sd 

S4 

ss 

82 

96 

100 

5 

Id 

15 

20 

25 

88 

35 

40 

45 

50 

55 

60 

65 

70 

75 

SO 

86 

90 

95 

loo 

105 

110 

115 

12o 

125 

6 

12 

18 

24 

80 

86 

4'2 

48 

64 

80 

66 

w 

T8 

84 

90 

96 

102 

10s 

114 

121' 

(26 

182 

188 

144 

150 

7 

14 

21 

28 

35 

42 

48 

56 

88 

To 

77 

s-l 

91 

98 

105 

112 

119 

1-26 

168 

(40 

147 

154 

161 

16s 

175 

8 

Itl 

24 

88 

40 

4^ 

56 

64 

T2 

BO 

88 

96 

104 

112 

120 

12S 

186 

144 

162 

160 

16s 

176 

1S4 

192 

200 

9 

[8 

27 

36 

45 

54 

68 

72 

M 

90 

98 

toe 

111 

126 

185 

144 

158 

1  02 

171 

1SI 

189 

19s 

207 

2  1C, 

225 

10    20 

30 

40 

60 

60 

70 

80 

M 

LOO 

110 

120 

180 

140 

150 

160 

170 

180 

190 

200 

210 

220 

280 

240 

250 

11    22 

88 

44 

55 

06 

77 

88 

99 

no 

121 

182 

148 

154 

165 

176 

181 

19s 

209 

22  ii 

281 

242 

268 

264 

275 

1*2    24 

86 

4- 

(in 

72 

M 

96 

108 

120 

132 

141 

166 

168 

180 

192 

204 

216 

22- 

241 

252 

264 

276 

23s 

300 

i:i   26 

55 

;>2 

DO 

rS 

51 

104 

1  1  * 

18d 

143  156 

168 

18-2 

195 

208 

221 

454 

247 

26i 

278 

286 

299 

U2 

325 

U    2- 

42 

66 

TO 

54 

98 

112 

126 

140 

15i'l68 

182 

196 

210 

224 

28s 

252 

26(1 

•2  si 

294 

!()s 

m 

$86 

350 

15    30 

45 

60 

75 

M 

lor. 

1-20 

185 

1  5i  i 

165180 

1  1*5 

•210 

225 

240 

255 

270 

286 

801 

815 

',80 

146 

-500 

375 

16 

82 

4s 

64 

80 

1)6 

112 

1-28 

144 

160 

176  192 

•20s 

224 

240 

256 

•272 

288 

804 

821 

386 

(62 

(68 

}S4 

400 

17 

84 

51 

68 

85 

109 

iisi 

186 

158 

170 

187'204 

281 

288 

255 

272 

2-9 

;o6 

828 

8K 

857 

174 

!91 

408 

425 

18    86 

64 

7'2 

90 

108 

126 

144 

162 

180 

198216 

'284 

252 

270 

288 

o()6 

524 

842 

8  ('.( 

87- 

596 

414 

432 

450 

19    88 

57 

76 

95 

114 

1:53 

152 

171 

190 

209  228 

24T 

266 

286 

804 

8-28 

U2 

861 

8-1 

:!99 

41S 

487 

456 

475 

20    40 

80 

80 

HIM 

120  140 

160 

180 

200 

220  240 

260 

•2-0 

800 

820 

840 

860 

;;su 

KM 

120 

440 

460 

4SO 

500 

^T 

42 

68 

>4 

lo,» 

126 

14, 

li>- 

[Sy 

2io 

231 

•2;V2 

2,:; 

21)  I 

81,> 

88C> 

85',' 

51§ 

899 

42( 

141 

46-2 

483 

504 

525 

22 

44 

66 

88 

110 

132 

ir,4 

176 

19- 

220 

242 

261 

286 

808 

880 

852 

874 

896 

4ls 

44( 

462 

4*4 

506 

528 

550 

28 

46 

96 

M 

115 

188 

161 

184 

2(i7 

280 

253 

2T( 

209 

822 

845 

868 

891 

414 

487 

46( 

488 

506 

529 

552 

575 

24 

48 

72 

M 

120 

144 

16s 

102 

216 

240 

264 

288 

812 

886 

860 

384 

408 

48'2 

456 

484 

504 

5-28 

55-2 

576 

600 

u 

so 

75 

10(1 

128 

160 

175 

•200 

•225 

•250 

275 

30( 

8-25 

861 

876 

400 

425 

450 

475 

5<;( 

5-25 

550 

575 

(500 

625 

1 

•2 

3 

4 

I 

0 

7 

8 

9 

10 

11 

12 

18 

14 

15 

1C 

17 

18 

lit 

20 

21 

22 

23 

24 

25 

77.  In  the  accompanying  table   take  the   multiplicand   in  the 
figures  arranged  horizontally  at  the  top  or  the  bottom,  and  the  multi- 
plier in  the  column  at  the  left ;    the  product  is  then  the  number 
under  or  above  the  multiplicand  and  opposite  the  multiplier. 

78.  Examples.     1.   Find  the  product  of  2418  x  7. 

2418  SOLUTION.     Write  the  multiplier  7  below  the  unit  figure  of 

7          the   multiplicand   as  shown  in  the  margin,  and  begin  at  the 

1fSQ2f»          right  to  multiply.     7  times  8  units  equals  56  units,  or  5  tens 

and  6  units.     Write  6  units  in  the  place  of  units,  and  reserve 

5  tens  to  add  to  the  product  of  tens.     7  times  1  ten  equals  7  tens,  and  adding 

the  5  tens  reserved  gives  12  tens,  or  1  hundred  and  2  tens.     Write  2  in  the  place 

of  tens,  and  reserve  1  to  add  to  the  product  of  hundreds.     7  times  4  hundreds 

equals  28  hundreds,  and  adding  the  1  hundred  reserved  gives  29  hundreds,  or 

2  thousands  and  9  hundreds.     Write  9  in  the  place  of  hundreds,  and  reserve 

2  to  add  to  the  product  of  thousands.     7  times  2  thousands  equals  14  thousands, 

and  adding  the  2  thousands  reserved  gives  16  thousands.     Write  this  entire  sum 

to  the  left  of  the  figures  already  written,  thus  completing  the  multiplication  and 

obtaining  as  a  result  16,926. 


38  SIMPLE  NUMBERS  [§§  78-83 

2.  Find  the  product  of  417  x  352. 

SOLUTION.  Write  the  multiplier  below  t,ne  multiplicand  in  the  same  unit 
order  from  the  right.  The  multiplier  is  composed  of  2  units,  6  tens,  and  3  hun- 
dreds, or  2  -f  50  -f  300.  Multiplying  the  -nultiplicand  by  2,  60,  and  300  respec- 
tively, and  adding  the  products,  the  results  are  as  follows  : 

(a)  FULL  PARTIAL  PRODUCTS      (6)  ABBREVIATED  PARTIAL  PRODUCTS 
417  417 

352  352 

834  Partial  Product  by  2  834 

20850  Partial  Product  by  60      2086 
125100  Partial  Product  by  300  1251  _ 
146784  Complete  Product          146784 

The  ciphers  at  the  right  of  the  partial  products  are  of  no  value  in  finding 
the  complete  product,  and  they  should  be  omitted  in  practice  as  shown  in  (6). 
Observe  that  the  first  right-hand  figure  of  each  partial  product  is  then  always 
directly  under  the  figure  of  the  multiplier  used. 

CONTRACTIONS  IN  MULTIPLICATION 

79.  To  multiply  any  number  by  10,  100,  1000,  etc. 

80.  Every  cipher  that  is  annexed  to  a  number  moves  each  digit 
one  place  to  the  left,  or  converts  units  into  tens,  tens  into  hundreds, 
and  so  on. 

81.  Hence  to  multiply  a  number  by  10, 100, 1000,  etc., 

To  one  factor  annex  as  many  ciphers  as  there  are  ciphers  in  the 
other  factor. 

82.  To  multiply  by  any  number  of  10's,  100's,  1000's,  etc. 

83.  Since  50  is  5  tens,  500  5  hundreds,  etc.,  to  multiply  a 
number  by  50,  1600,  7000,  etc., 

Omit  the  ciphers  on  the  right  of  the  factors,  multiply  the 
remaining  part  of  the  multiplicand  by  the  remaining  part 
of  the  multiplier,  and  to  the  product  thus  obtained  annex 
the  ciphers  omitted. 

ORAL  EXERCISE 

By  inspection,  find  the  products  of: 

1.  34,000x600.               5.   710x6000.  9.  716x20. 

9.  216,000x300.             6.   72x3000.  10.  160x20. 

3.  5400x70.                    7.   714,000x200.  11.  805x2000. 

4.  5120x2000.                8.   28,000x20.  12.  43,007x100 


§§  84-86]  MULTIPLICATION  39 

84.  To  multiply  any  number  by  11. 

85.  Examples.     1,   Find  the  product  of  72  x  11. 

(a;  FULL  OPERATION  (6)  CONTRACTED  OPERATION 

72  Write  2  in  the  place  of  the  units  of  the 

11  product.  2  +  7  =  9.  Write  9  in  the  place  of 

72  the  tens  of  the  product.  Bring  down  7  for  the 

72  place  of  the  hundreds  of  the  product. 

792  The  completed  product  is  therefore  792. 

SOLUTION.     By  glancing  at  (a)  it  will  be  observed  that : 

1.  The  partial  product  by  the  units  of  the  multiplier  contains  the  same 
figures  as  the  partial  product  by  the  tens. 

2.  The  first  figure  in  the  partial  product  by  tens  of  the  multiplier  falls  under 
the  second  figure  of  the  partial  product  by  units. 

Making  practical  use  of  these  observations,  we  have  the  abbreviated  opera- 
tion as  in  (6). 

2.  Find  the  product  of  89  X  11. 

SOLUTION.  Write  9  as  the  first  figure  of  the  product.  9  +  8  =  17.  Write  7 
as  the  second  figure  of  the  product,  and  carry  1.  8+1=9.  Write  9  as  the  third 
figure  of  the  product,  thus  completing  the  multiplication  and  obtaining  as  a  re- 
sult 979. 

3.  Find  the  product  of  195  X  11. 

SOLUTION.  Write  5  as  the  first  figure  in  the  product.  6  +9  =  14.  Write  4 
as  the  second  figure  in  the  product,  and  carry  1.  9  +  1  +  1  =  11.  Write  1  as 
the  third  figure  in  the  product,  and  carry  1.  1  +  1  =  2.  Write  2  as  the  fourth 
figure  in  the  product,  thus  completing  the  multiplication  and  obtaining  as  a  re- 
sult 2145. 

86.  Hence  the  rule : 

Write  as  the  first  figure  of  the  product  the  first  figure  of  the 
multiplicand. 

Beginning  at  the  right  of  the  multiplicand  add  the  units 
and  tens,  the  tens  and  hundreds,  the  hundreds  and  thousands, 
and  so  on. 

Finally,  bring  down  tJie  left-hand  figure  of  the  multipli- 
cand as  the  left-hand  figure  of  the  product.  Carry  when 
necessary. 


40  SIMPLE  NUMBERS  [§§86-88 

ORAL  EXERCISE 
By  inspection,  find  the  products  of : 


1. 

62  X 

11. 

11. 

37  X 

11. 

21. 

52  x 

11. 

81. 

225  x  11. 

2. 

51  X 

11. 

12. 

11  x 

43. 

22. 

85  x 

11. 

82. 

11  X 

428. 

S. 

11  X 

71. 

18. 

59  x 

11. 

23. 

11  X 

93. 

S3. 

11  x 

927. 

4^ 

45  X 

11. 

14. 

11  x 

78. 

24. 

11  x 

68. 

0  / 

34- 

11  x 

728. 

6. 

11  X 

29. 

15. 

81  x 

11. 

25. 

121  X  11. 

85. 

726  x  11. 

6. 

11  X 

75. 

16. 

21  x 

11. 

26.  132x11. 

86. 

487  x  11. 

7. 

11  X 

84. 

17. 

24  x 

11. 

27. 

11  X 

141. 

87. 

1926 

xll. 

8. 

11  X 

91. 

18. 

11  x 

26. 

28. 

11  X 

164. 

88. 

11  X 

1726. 

9. 

86  x 

11. 

19. 

11  X 

34. 

29. 

11  X 

214. 

89. 

11  X 

2814. 

10.  32x11.       m  11x47.       SO.  216x11.       40.   5419x11. 

87.  To  multiply  by  any  number  of  ll's ;  as,  22,  33,  etc. 

88.  Examples.    1.   Multiply  24  by  22. 

SOLUTION.  4x2=8.  Write  8  as  the  first  figure  of  the  product. 
6(4  +  2)  x  2  =  12.  Write  2  as  the  second  figure  of  the  product,  and  carry  1. 
2x2  +  1  =  5.  Write  5  as  the  third  figure  of  the  product,  thus  completing  the 
multiplication  and  obtaining  as  a  result  528. 

2.  Find  the  product  of  121  x  77. 

SOLUTION.  Write  7  as  the  first  figure  of  the  product.  3(1  +  2)  x  7  =  21. 
Write  1  as  the  second  figure  of  the  product  and  carry  2.  3(2  +  1)  x  7  +  2  =  23. 
Write  3  as  the  third  figure  of  the  product  and  carry  2.  1x7  +  2  =  9.  Write  9 
as  the  fourth  figure  of  the  product,  thus  completing  the  multiplication,  and 
obtaining  as  a  result  9317. 

It  should  be  observed  that  this  method  is  practically  the  same  as  the  method 
of  multiplying  by  11. 

WRITTEN  EXERCISE 

Find  the  products  of : 

1.  21x22.  5.  33x425.  9.   125x88.  13.  146x55. 

2.  44x36.  6.  42x33.  10.   22x1214.  14.  22x216. 
8.  12x22.  7.  66x71.  11.   44x74.  15.  151x44. 
4.  66x215.  8.  214x33.  12.  25x88.  16.  66x125 


§§  89-00]  MULTIPLICATION  41 

89.  To  multiply  by  any  number,  one  part  of  which  is  contained  a 
certain  number  of  times  in  another  part. 

90.  Examples.    L  Find  the  product  of  254  x  357. 

SOLUTION.     Observe  that  the  tens  and  hundreds  (35)  of  the 

357     multiplier  make  a  number  five  times  the  units  (7).     Multiply  254 

1778     by  7  in  the  usual  manner.     Multiply  the  resulting  partial  product 

8890       by  5  (the  Partial  product  by  7  multiplied  by  6  equals  the  partial 

90678     product  by  35),  and  complete  the  multiplication  by  adding. 

8.  Find  the  product  of  12,121  x  12,816. 

12121 

12816  SOLUTION.     Multiply  12,121  by  16,  obtaining  193,936. 

1QoQqfi       Multiply  193,936  by  8  (128  is  just  8  times  16),  and  obtain 
1,651,488.      Add   the    two   partial   products   and    obtain 
165,342,736,  thus  completing  the  multiplication. 
155342736 

WRITTEN  EXERCISE 

1.  If  a  page  contains  1864  ems,  how  many  ems  in  a  book  of 
794  pages  ? 

2.  If  735  men  can  dig  a  canal  in  9328  days,  how  many  men 
would  be  required  to  complete  the  work  in  1  day  ? 

S.   How  many  links  in  639  chains,  each  chain  having  8471  links  ? 

4.  The  Boston  "  Boot  Maker "  will  enable  a  workman  to  make 
324  pairs  of  boots  daily.    How  many  can  be  made  with  this  machine 
in  328  days  ? 

5.  What  must  be  paid  for  grading  a  railroad  1809  miles  long  at 
$1288  a  mile? 

6.  What  will  248  acres  of  land  cost  at  $  217  per  acre  ? 

7.  If  the  circulation  of  the  city  library  is  27,126  books  daily, 
how  much  would  it  be  in  168  days  ? 

8.  A  dray  horse  can  draw  10  loads,  of  1569  pounds  each,  per  day. 
How  many  pounds  can  749  horses  draw  in  1  day  at  the  same  rate  ? 

9.  A  barrel  of  flour  weighs  196  pounds.     What  is  the  weight  of 
639  barrels  ? 

10.  A  railway  is  1449  miles  in  length,  and  was  completed  at  an 
average  cost  of  $106,775  per  mile.  What  was  the  total  cost  of 
constructing  it  ? 


42  SIMPLE   NUMBERS  [§§  91-93 

91.  Cross  Multiplication.     The  possibilities  of  what  is  known  as 
cross  multiplication  are  almost  without  end.     The  method  is  par- 
ticularly helpful  in  making  mental  extensions  on  invoices.     By  it 
the  product  of  any  two  numbers  of  two  figures  each  may  be  ascer- 
tained mentally,  and  the  product  of  any  number  by  any  other  num- 
ber of  two  figures  may  be  obtained  by  simply  writing  the  completed 
product.      By  intelligent,  persistent  practice  any  number  may  be 
multiplied  by  any  other  number  of  three  or  four  figures  without 
writing  any  of  the  partial  products  that  are  ordinarily  written  in 
multiplying  one  number  by  another. 

92.  Examples.     1.  Find  the  product  of  74  x  23. 

SOLUTION.    4  x  3  =  12.    Write  2  as  the  first  figure  of  the  product 

74       and  carry  1.     7x3  +  1  (carried)  +  8  (4  x  2)  =  30.     Write  0  as  the 

23       second  figure  of  the  product  and  carry  3.    7x2  +  3  (carried)  =  17. 

1702       Write  17  to  the  left  of  the  figures  already  written  in  the  product,  thus 

completing  the  multiplication  and  obtaining  a  product  of  1702. 

&   Find  the  product  of  124  x  62. 

.  ^  £ ,  SOLUTION.    4x2  =  8.    Write  8  as  the  first  figure  of  the  product. 

2  x  2  +  24  (4  x  6)  =  28.     Write  8  as  the  second  figure  of  the  prod- 
uct and  carry  2.     1  x  2  +  12  (2  x  6)  +  2  (carried)  =  16.     Write  6 


7688       as  the  third  figure  of  the  product  and  carry  1.      1x6  +  1  (car- 
ried) =  7.     Write  7  as  the  fourth  figure  of  the  product,  thus  com- 
pleting the  multiplication  and  obtaining  a  product  of  7688. 

3.  Find  the  product  of  2146  x  32. 

SOLUTION.     6  x  2  =  12.      Write    2    and    carry    1.      4x2  +  1 
(carried)  +  18(6  x  3)  =  27.       Write   7    and   carry   2.       1x2  +  2 

^      (carried)  +  12(4  x  3)  =  16.       Write   6  and    carry    1.      2x2  +  1 

68672     (carried)  +  3(1  x  3)  =  8.      Write  8.      2x3=6.      Write    6,   thus 
completing  the  multiplication  and  obtaining  a  product  of  68672. 

4.  Find  the  product  of  214  x  236. 

SOLUTION.     4x6  =  24.     Write  4  and  carry  2.     1x6  +  2  + 

12(4  x  3)  =  20.      Write  0  and  carry  2.      2  x  6  +  2  +  3(1  x  3)  + 

236     8(4  x  2)  =  25.     Write  5  and  carry  2.     2x3  +  2  +  2(1x2)  =  10. 

50504     Write  0  and  carry  1.      2x2+1=5.      Write  5,  thus  completing 

the  multiplication  and  obtaining  a  product  of  50">04- 

93.  The  method  of  cross  multiplication  can  hardly  be  covered 
by  a  set  rule,  since  it  includes  such  a  wide  range  of  numbers.  In 
attempting  to  make  practical  application  of  this  method  the  "follow- 
ing principles  are  important. 


§§  93-95]  MULTIPLICATION  43 

1.  Units  multiplied  by  units  equal  the  units  of  the  product. 

2.  Tens  multiplied  by  units,  plus  units  multiplied  by  tens  equal 
the  tens  of  the  product. 

3.  Hundreds  multiplied  by  units  plus  tens  multiplied  by  tens 
plus  units  multiplied  by  hundreds  equal  hundreds  of  the  product. 

4.  Thousands  multiplied  by  units  plus  hundreds  multiplied  by 
tens   plus  tens   multiplied   by   hundreds  plus  units  multiplied  by 
thousands  equal  thousands  of  the  product. 

WRITTEN  EXERCISE 

Find  the  product  in  each  of  the  following  problems.     Do  not  use 
pen  or  pencil  except  to  write  the  product. 


1. 

2. 
3. 

* 

5. 
6. 

24  x  32. 
41  x  35. 
2115  x  32. 
127  x  23. 
53  x  42. 
2144  x  36. 

7. 
8. 
9. 
10. 
11. 
12. 

1121  x  42. 
116  x  45. 
47  x  26. 
37  x  48. 
1174  x  26. 
181  x  59. 

13. 
14. 
15. 
16. 
17. 
18. 

125  x  34. 
36  x  58. 
1215  x  57. 
2125  x  64. 
164  x  32. 
172  x  27. 

19. 
20. 

21. 

22. 
c><s> 

KO. 
24. 

34  x  51. 
1217  x  42. 
241  x  36. 
142  x  28. 
45  x  91. 
3215  x  42. 

94.  To  multiply  any  number  by  the  numbers  from  101  to  109 
inclusive. 

95.  Examples.    1.   Find  the  product  of  64  x  102. 

64 
..  ~o  SOLUTION.    64  x  2  =  128.     Write  28  and  carry  1.    64  x  1  + 1  = 

— —     65.     Write  65  to  complete  the  product. 
6528 

2.   Find  the  product  of  215  x  102. 

215  SOLUTION.     15  x  2  =  30.     Write  30  for  the  first  figures  of  the 

102     product.     2x2  +  5  =  9.     Write  9  as  the  third  figure  of  the  product. 
21930     21  x  1  =  21.     Write  21  to  complete  the  product. 

S.   Find  the  product  of  2265  x  104. 

2265  SOLUTION.     5x4  =  20.    Write  0  and  carry  2.    6  x  4  +  2  =  26. 

104      Write  6  and  carry  2.     2  x  4  +  2  +  5  =  15.     Write  5  and  carry  1 
235560      2x4  +  1+6  =  15.     Write  5  and  carry  1.     22  x  1  +  1  =  23. 
Write  23  to  complete  the  product. 

NOTE.    The  above  method  of  multiplication  may  be  used  to  advantage  in 
billing  where  the  price  is  $1.02  and  $1.03,  etc. 


44  SIMPLE    .NUMBERS  Cf§  96-100 

ORAL  EXERCISE 

By  inspection,  find  the  product  of : 

1.  32  x  102.      4.     53  x  105.      7.     58  x  102.     10.   105  x  94. 

2.  103  x  47.        5.     72  x  106.      8.    104  x  32.       11.     71  x  102. 

3.  39  x  104.      6.   114  x  105.      9.  106  x  58.      12.     88  x  101. 

96.  To  multiply  by  any  number  of  three  figures,  the  tens  of  which 
is  a  cipher. 

97.  Examples.     1.  Find  the  product  of  126  X  302. 

SOLUTION.     26  x  2  =  52.     Write  52  as  the  first  two  figures  of 


the  product.    1x2  +  6x3  =  20.     Write  0  as  the  third  figure 
**          of  the  product  and  carry  2.     12  x  3  +  2  =  38.     Write  38  to  com- 


38052     piete  tbe  product. 

2.  Find  the  product  of  1215  x  304 

SOLUTION.     15  x  4  =  60.    Write  60  as  the  first  two  figures 

1215       of  the  product.     2x4  +  5x3  =  23.     Write  3  as  the  third  figure 

304       of  the  product,  and  carry  2.     1x4  +  2  (carried)  +  1x3  =  9. 


369360       Write  9  as  the  fourth  figure  of  the  product.    12  x  3  =  36.    Write 

36  to  complete  the  product. 

It  will  be  observed  from  the  above  solutions  that  this  method  is  practically 
the  same  as  94. 

WRITTEN  EXERCISE 

Find  the  product  in  each  of  the  following  problems  without  using 
pen  or  pencil  except  to  write  the  figures  of  the  completed  product. 
1.   121  x  202.       5.   305  x  408.         9.  413  x  301.       13.   123  x  407 
9.   116  x  403.       6.  431  x  309.      10.  365  x  308.       14.  218  x  905 

3.  151  x  304.       7.   918  x  201.      11.  413  x  503.       15.   721  x  801 

4.  165  x  405.       8.   725  x  402.      12.  936  x  405.       16.  718  x  203 

98.  To  square  any  number  of  two  figures. 

99.  To  square  any  number  of  two  figures  the  method  of  cross 
multiplication  may  be  used,  or  the  work  may  be  further  contracted 
as  shown  in  the  following  example. 

100.    Example.     Square  72. 

_rt  SOLUTION.    2x2  =  4.    Write  4  as  the  first  figure  of  the  product. 

14  (7  +  7  )  x  2  =  28.     Write  8  as  the  second  figure  of  the  product,  and 
*"     carry  2.     7x7  +  2  =  51.     Write  51  to  complete  the  product  ;  or, 


5184  2x2  =  4.     Write  4  as  the  first  figure  of  the  product.    4  (2  +  2} 

x  7  =  28.     Write  8  as  the  second  figure  of  the  product,  and  carry  2 
7  x  7  +  2  =  61.     Write  51  to  complete  the  product. 


§  101]  MULTIPLICATION  45 

101.    Therefore  the  following  rule : 

Multiply  the  units  of  the  multiplicand  by  the  units  of  the 
multiplier  and  write  the  result  in  the  product t  carrying  as 
usual. 

Add  the  tens  in  the  multiplier  to  the  tens  of  the  multi- 
plicand and  multiply  by  the  units  of  the  multiplier;  or,  add 
the  units  of  the  multiplier  to  the  units  of  the  multiplicand 
and  multiply  the  sum  by  the  tens  of  the  multiplier.  Write 
the  result  in  the  product,  carrying  as  usual. 

Multiply  the  tens  of  the  multiplicand  by  the  tens  of  the 
multiplier,  and  write  the  full  result  in  the  product. 

WRITTEN  EXERCISE 

1.  Find  the  sum  of  the  squares  of  the  following  numbers. 

24,    36,    32,    34,    67,    84,     92,     76,    89,    47,    39,    38,    43, 
56,    75,    88,    95,    83,    94,    71,    29,    44,    59,    65,    73. 

2.  Square  the  numbers  written  below,  writing  the  results  hori- 
zontally ;  then  find  the  sum  of  -the  squares. 

37,    48,    68,    62,    98,    26,    27. 

8  Square  the  numbers  written  below,  and  from  the  sum  of  the 
first  five  squares  subtract  the  sum  of  the  second  five. 

63,    61,    49,    76,    81,    41,    33,    23,    35,    37. 

ORAL  REVIEW 
By  inspection,  find  the  cost  of  each  of  the  following  items : 

1.  215  pounds  of  butter  at  22^;  102  pounds  at  28  £ 

2.  104  bushels  of  garden  corn  at  $  1.75;  204  bushels  at  52  £ 
8.   125  bushels  of  wheat  at  $  1.02 ;   115  bushels  at  $1.05. 

4.  102  barrels  of  apples  at  $1.48;  103  barrels  at  $1.67. 

5.  33  bushels  of  pears  at  $1.39;  55  bushels  at  $1.15. 

6.  65  bushels  of  potatoes  at  65  ^ ;  58  bushels  at  52  #. 

7.  44  baskets  of  peaches  at  $1.23;  52  baskets  at  87  £ 

8.  64  barrels  of  flour  at  $5.15;  33  barrels  at  $6.50. 

9.  26  bags  of  bran  at  $1.03;  32  bags  at  $1.05. 


46  SIMPLE   NUMBERS  [§  101 

10.  125  pounds  of  coffee  at  22^;  164  pounds  at  33^. 

11.  56  gallons  of  molasses  at  36^;  84  gallons  at  34^. 

12.  43  pounds  of  chocolate  at  43  ^ ;  52  pounds  at  52  ^. 
18.  23  sacks  of  pancake  flour  at  42^ ;  45  sacks  at  45^. 
14.  62  boxes  of  ice  cream  salt  at  62^;  53  boxes  at  51^. 

WRITTEN  REVIEW 

7.  A  manufacturer  sold  171  corn  shellers  at  $23  each.    How 
much  did  he  receive  for  them  ? 

2.   There  are  5280  feet  in  a  mile.     How  many  feet  are  there  in 
104  miles  ? 

8.  What  will  462  barrels  of  petroleum  cost  at  $  1.08  per  barrel  ? 

4.  In  freighting,  lime  and  flour  are  each  estimated  to  weigh  200 
pounds  per  barrel ;  pork  and  beef,  each  32  pounds ;  apples  and  pota- 
toes, each  150  pounds ;  cider,  whisky,  and  vinegar,  each  150  pounds. 
What  will  be  the  weight  of  the  freight  in  a  car  containing  22  barrels 
of  each  of  these  products  ? 

5.  If  a  bushel  of  barley  weighs  48  pounds,  of  clover  seed  60 
pounds,  of  flax  seed  55  pounds,  of  beans  60  pounds,  of  buckwheat  48 
pounds,  of  rye  56  pounds,  of  corn  56  pounds,  of  oats  32  pounds,  of 
potatoes  60  pounds,  of  timothy  seed  45  pounds,  of  wheat  60  pounds, 
what  will  be  the  total  weight  of  77  bushels  of  each  product  ? 

6.  A  man  rented  a  farm  of  132  acres  of  grain  land,  76  acres  of 
pasture  land,  and  45  acres  of  meadow  land ;  paying  for  the  grain  land 
$7  per  acre,  for  the  pasture  land  $4  per  acre,  and  for  the  meadow 
land  $  11  per  acre.     He  produced  61  bushels  of  oats  per  acre  on  45 
acres,  32  bushels  of  barley  per  acre  on  30  acres,  75  bushels  of  corn 
per  acre  on  15  acres,  150  bushels  of  potatoes  per  acre  on  9  acres,  28 
bushels  of  buckwheat  per  acre  on  20  acres,  and  24  bushels  of  beans 
per  acre  on  the  remainder  of  the  grain  ground.    He  relet  the  pasture 
land  for  $  200,  and  on  the  meadow  cut  2  tons  per  acre  of  hay,  worth 
$13  per  ton.     If  he  paid  $695  for  labor  and  $467  for  other  ex- 
penses, and  sold  the  oats  at  26  ^  per  bushel,  the  barley  at  65  ^,  the 
corn  at  40^,  the  potatoes  at  33^,  the  buckwheat  at  60^,  and  the 
beans  at  $  2,  did  he  gain  or  lose,  and  how  much  ? 


§§  102-111]  DIVISION  47 

DIVISION 

102.  Division  is  the  process  of  finding  how  many  times  one 
number  is  contained  in  another. 

103.  The  dividend  is  the  number  to  be  divided. 

104.  The  divisor  is  the  number  by  which  the  dividend  is  to  be 
divided. 

105.  The  quotient  is  the  result  obtained  by  division. 

106.  Division  is  exact  when  all  the  dividend  is  divided  and  the 
quotient  is  an  integer. 

107.  The  remainder  is  the  part  left  undivided  when  the  division 
is  not  exact. 

108.  The  sign  -*-  signifies  division  and  is  read  divided  by. 

Thus  24  -7-  8  =  3  is  read  24  divided  by  8  equals  3. 

The  dividend  in  division  corresponds  to  the  product  in  multiplication,  and 
the  divisor  and  quotient  to  the  multiplicand  and  multiplier,  respectively. 

109.  General  Principles.     1.   Multiplying  the  dividend  or  divid- 
ing the  divisor  multiplies  the  quotient. 

2.  Dividing  the  dividend  or  multiplying  the  divisor  divides  the 
quotient. 

3.  Multiplying  or  dividing  both  the  dividend  and  divisor  by  the 
same  number  does  not  change  the  quotient. 

4.  When  the  divisor  and  dividend  are  like  numbers,  the  quotient 
is  an  abstract  number. 

5.  When  the  divisor  is  an  abstract  number,  the  dividend  and 
quotient  are  like  numbers. 

110.  When  the  divisor  is  so  small  that  the  division  may  be  per- 
formed mentally,  the  process  is  called  short  division. 

111.  Example.    Divide  3713  by  8. 

SOLUTION.     Write  the  divisor  at  the  left  of  the  dividend  with 
a  curved  line  between  them. 


8)3713  8  is  not  contained  in  3  thousands,  therefore  divide  37  hun- 

dreds by  8.     8  is  contained   in  37  hundreds  4  times  with  a 

remainder  5.     Write  4  in  the  quotient  in  the  place  of  hundreds.    Reducing  5 

hundreds  to  tens  and  adding  the  one  ten  of  the  dividend,  the  result  is  51.     8  is 

contained  6  times  in  51  with  a  remainder  8.     Write  6  in  the  quotient  in  the 


48  SIMPLE   NUMBERS  [§§  111-113 

place  of  tens.  Reducing  the  3  tens  to  units  and  adding  the  3  units  of  the  divi- 
dend, the  result  is  33  units.  8  is  contained  in  33  units  4  times  with  a  remainder 
1.  Write  4  in  the  place  of  units  and  place  the  remainder  over  the  divisor  with 
a  line  between  ;  thus,  $.  The  complete  quotient  is  464£. 

112.  When  the  divisor  is  so  large  that  each  step  in  the  division 
must  be  Written,  the  process  is  called  long  division. 

113.  Example.     Divide  5207  by  98. 

SOLUTION.     Since  98  is  more  than  52,  it  is  necessary  to  take 
__  520  for  the  first  partial  dividend.     The  nearest  number  of  tens 

98)5207          represented  by  the  divisor  is  10 ;  take  10,  therefore,  for  the 
490  trial  divisor.    The  number  of  tens  in  the  partial  dividend  is 

307          52.     10  is  contained  5  times  in  52.     Write  5  in  the  quotient 
294         over  the  right-hand  figure  of  the  partial  dividend  as  shown  in 
~~13         the  margin.     Multiplying  the  exact  divisor  by  5  and  subtract- 
ing the  product  from  the  partial  dividend,  the  remainder  is  30, 
to  which  annex  the  7  units  of  the  dividend,  and  the  second  partial  dividend  is 
307.     The  nearest  number  of  tens  represented  by  the  second  partial  dividend  is 
31,  which  will  contain  10  3  times.     Write  3  in  the  quotient.    Multiplying  the 
exact  divisor  by  3  and  subtracting  the  product,  the  remainder  is  13,  to  be  written 
in  the  form  of  a  fraction  and  annexed  to  the  quotient.     The  complete  quotient 
is  53H- 

ORAL  EXERCISE 

1.  The  quotient  is  61.     If  the  dividend  and  divisor  were  each 
multiplied  by  4,  what  would  the  quotient  be  ? 

2.  The  quotient  is  53.     If  the  dividend  and  divisor  were  each 
divided  by  3,  what  would  the  quotient  be? 

3.  If  the  divisor  were  4  times  what  it  is,  the  quotient  would  be 
1606.     What  is  the  quotient  ? 

4.  The  quotient  of  one  number  divided  by  another  is  12.     What 
would  the  quotient  be  if  the  divisor  were  multiplied  by  3  ?    divided 
by  3? 

5.  How  many  15's  must  we  add  together  to  get  4590  ? 

WRITTEN  EXERCISE 

1.  $21,735  was  received  from  the  sale  of  a  farm  at  $35  an  acre. 
How  many  acres  did  the  farm  contain  ? 

2.  What  number  must  be  added  to  21,786  that  it  may  be  exactly 
divisible  by  168  ? 


§§  113-115]  DIVISION  49 

8.   The  remainder  is  14,  the  quotient  5041,  and  the  divisor  15, 
What  is  the  dividend  ? 

4.  The  remainder  is  7,  the  quotient  19,023,  and  the  dividend 
247,306.     What  is  the  divisor  ? 

5.  If  8  men  can  do  a  piece  of  work  in  24  days,  in  how  many 
days  can  12  men  do  the  same  work  ? 

6.  I  sell  my  village  home  for  $  3250,  my  store  for  $5000,  my 
stock  of  goods  for  $11,250,  receiving  in  part  payment  $8775,  and 
for  the  remainder,  Iowa  prairie  land  at  $15  per  acre.     How  many 
acres  should  I  receive  ? 

7.  If  there  are  128  cubic  feet  in  1  cord,  how  many  cords  in 
141,492  cubic  feet? 

8.  If  93  be  added  to  a  certain  number,  it  will  contain  648 
twenty-five  times.     What  is  the  number? 

9.  What  number  must  be  subtracted  from  3476  that  it  may  be 
exactly  divisible  by  155  ? 

10.  A  man  bought  490  acres  of  land  at  $  40  an  acre  and  after 
paying  $2900  for  improvements  sold  it  for  $25,000.  Did  he  gain 
or  lose,  and  how  much  ? 

CONTRACTIONS  IN  DIVISION 

114.  To  divide  any  number  by  10,  100,  1000,  etc. 

115.  By  the  decimal  system  of  notation  numbers  increase  in 
value  from  right  to  left  and  decrease  from  left  to  right  in  a  tenfold 
ratio ;  hence  to  divide  a  number  by  10,  100,  1000,  etc. : 

From,  the  right  in  the  dividend  point  off  as  many  places 
as  the  divisor  contains  ciphers. 

The  figures  so  cut  off  express  the  remainder,  to  be  written 
in  fractional  form. 

ORAL  EXERCISE 

By  inspection,  find  the  complete  quotients  of: 

L   759  -=-10.  4.   4997-- 10000.          7.   297249-*- 10. 

2.  7527929-^1000.         5.   75627 -j- 10.  8.   759^-10000. 

3.  29-5-100.  6.   8967^100.  9.   4627490 -=- 1000. 


50  SIMPLE   NUMBERS  [§§116-122 

116.  To  divide  by  any  number  of  10's,  100's,  1000's,  etc. 

117.  Example.     Find  the  quotient  of  14,131  -s-  4000. 

SOLUTION.     Mark  off  as  many  figures  in  the  dividend  as 

there  are  ciphers  in  the  divisor,  thus  dividing  by  1000.     The 

**Tffinr      first  quotient  is  then  14  with  a  first  remainder  131.     Dividing 

4|000)14|131        14  by  4  gives  3  as  the  final  quotient  with  a  second  remainder  2. 

12  Multiplying  the  second  remainder  by  1000  to  obtain  its  true 

~2131         value  and  adding  the  first  remainder,  the  result  is  the  true 

4QQQ         remainder,  2131,  to  be  written  in  fractional  form  and  placed 

by  the  side  of  the  integral  quotient.    The  completed  quotient 

is  then  Sfflft. 

ORAL  EXERCISE 
By  inspection,  find  the  complete  quotients  of: 

1.  1627 -f- 400.  8.   762179 -=-190000.        15.   7849-5-260. 

2.  571119 -j- 19000.          9.   51295 -j- 17000.  16.  8479-5-280 

5.  48887 --40.  10.   6439-- 160.  17.   9579-- 190. 
4.   6427 --80.                 11.   19279 -- 160.  18.   125265 -- 250. 

6.  9687-- 120.  12.  15597-5-5000.  19.   16219 -v- 4000. 

6.  7879-5-390.  IS.  21259 -v- 700.  20.  39379-5-13000. 

7.  19249-5-4000.  14.   72899-^-24000.  21.  4179-5-1200. 

PROPERTIES  OF  NUMBERS 

118.  Properties  of  numbers  are  those  qualities  which  belong  to 
and  are  inseparable  from  them. 

119.  All  integral  numbers  are:  (1)  odd  or  even;  (2)  prime  or 
composite. 

120..  An  odd  number  is  a  number  that  cannot  be  exactly  divided 
by  2  ;  as,  5,  9,  23. 

121.  An  even  number  is  a  number  that  can  be  exactly  divided  by 
2  ;  as,  6,  8,  44. 

122.  Factors  of  numbers  are  those  numbers  the  continued  product 
of  which  will  produce  the  number. 


§§  ^  23-131]  PROPERTIES   OF  NUMBERS  51 

123.  A  prime  number  is  a  number  that  cannot  be  resolved  into 
two  or  more  factors  ;  or, 

it  is  a  number  that  has  no  integral  factors  except  unity  and  itself. 
Thus,  23,  59,  11,  and  13  are  prime  numbers.     2  is  the  only  even  number 
that  is  prime. 

124.  A  composite  number  is  a  number  that  can  be  resolved  into 
two  or  more  factors ;  or, 

it  is  a  number  which  is  the  product  of  twro  or  more  integral 
factors. 

125.  A  prime  factor  is  a  prime  number  used  as  a  factor. 

126.  A  composite  factor  is  a  composite  number  used  as  a  factor. 

127.  An  exact  divisor  of  a  number  is  any  integral  factor  of  that 
number. 

128.  A  common  divisor  of  two  or  more  numbers  is  any  exact 
divisor  of  those  numbers. 

129.  The  greatest  common  divisor  of  two  or  more  numbers  is  the 
greatest  exact  divisor  common  to  those  numbers. 

130.  Numbers  having  no  common  divisor  or  factor  are  said  to  be 
relatively  prime. 

131.  Tests  of  Divisibility.  *  In  arithmetical  computations  it  is  fre- 
quently necessary  to  determine  whether  one  number  is  divisible  by 
another  or  not.     In  dividing  numbers  the  following  tests  of  divisi- 
bility will  be  found  helpful. 

1.  When  a  number  is  even,  it  is  divisible  by  2. 

2.  When  the  sum  of  the  digits  of  any  number  is  divisible  by  3 
or  9,  the  whole  number  is  divisible  by  3  or  9. 

3.  When  the   right-hand  figure  of  any  number  is  5  or  0,  the 
whole  number  is  divisible  by  5. 

4.  When  the  right-hand  figure  is  0,  the  whole  number  is  divis- 
ible by  10. 

5.  When  the  number  expressed  by  the  two  right-hand  figures 
of  a  number  is  divisible  by  4,  the  whole  number  is  divisible  by  4. 

6.  When  a  number  is  even  and  divisible  by  3,  it  is  also  divisible 
by  6. 

7.  When  the  number  expressed  by  the  three  right-hand  figures 
of  a  number  is  divisible  by  8,  the  whole  number  is  divisible  by  8. 


52  SIMPLE   NUMBERS  [§§  132-ldG 

132.  A  multiple  of  a  number  is  one  or  more  times  the  number;  or, 
it  is  that  product  of  which  the  given  number  is  an  exact  divisor. 

133.  A  common  multiple  of  two  or  more  numbers  is  that  product 
of  which  the  given  numbers  is  an  exact  divisor. 

134.  The  least  common  multiple  of  two  or  more  numbers  is  the 
least  product  of  which  each  of  the  given  numbers  is  an  exact  divisor. 

FACTORING 

135.  Factoring  is  the  process  of  separating  or  dissolving  a  com- 
posite number  into  factors. 

136.  General  Principles.     1.   Any  composite  number  is  divisible 
by  each  of  its  several  factors  successively. 

2.   A  composite  number  is  equal  to  the  product  of  all  its  prime 
factors. 

137.  To  find  the  prime  factors  of  a  composite  number. 

138.  Example.    Find  the  prime  factors  of  4290. 

4290  SOLUTION.    The  given  number  ends  with  a  0,  hence  is  exactly 


2 

3 

11 


gpjg  divisible  by  5.  Dividing  by  5,  the  quotient  858  is  obtained.  858,  being 
an  even  number,  is  exactly  divisible  by  2.  Dividing  by  2,  the  quotient 
429  is  obtained.  The  sum  of  the  digits  in  429  is  divisible  by  3 ;  there- 
fore the  whole  number  is  divisible  by  3.  Dividing  by  3,  the  quotient 


143 


143  is  obtained.  143  is  exactly  divisible  by  11.  Dividing  byll,  the 
quotient  13  is  obtained.  The  several  divisors,  5,  2,  3,  11,  and  the  last  quotient, 
13,  are  the  prime  factors  required. 

139.    Therefore  the  following  rule  : 

Divide  the  given  number  by  any  prime  factor. 

Divide  the  successive  quotients  in  tlw  same  manner  until 
a  quotient  that  is  prime  is  obtained. 

The  several  divisors  and  the  last  quotient  are  the  prime 
factors  required. 

WRITTEN  EXERCISE 

Find  the  prime  factors  of  : 

1.  144.          8.   924.          6.  135.  7.   1575.  9.  951. 

2.  124.          4.  289.          6.   25785.          8.   252.  10.   1527. 


§§  140-144]  PROPERTIES   OF   NUMBERS  53 

GREATEST  COMMON  DIVISOR 

140.  To  find  the  greatest  common  divisor  of  two  or  more  numbers. 

141.  Example.     Find  the  greatest  common  divisor  of  42,  66, 

and  84. 

SOLUTION.     Arrange  the  numbers  as  shown  in  Jtlie 
margin.     The  first  prime  factor  that  will  divide  all  the 


numbers  is  2.      Divide  by  2,  obtaining  21,  33,  and  42 


7  — 11  —  14         as  the  quotients.     The  only  prime  factor  that  is  common 
to  these  numbers  is  3.    Divide  by  3,  obtaining  as  the 

quotient  7, 11,  and  14.  There  is  no  factor  common  to  all  of  these  numbers.  Since 
2  will  divide  all  the  given  numbers,  and  3  will  divide  the  resulting  quotients,  the 
product  of  2  x  3,  or  6,  is  the  greatest  common  divisor  of  42,  66,  and  84. 

142.  Hence  the  following  rule : 

Write  the  numbers  in  a  horizontal  line,  separating  them 
by  dashes. 

Divide  by  any  prime  number  that  will  divide  all  the  given 
numbers  without  a  remainder,  writing  the  quotients  in  a 
line  below.  Continue  the  process  until  the  quotients  have  no 
common  factor. 

Multiply  together  the  several  divisors,  and  the  result  is  the 
greatest  common  divisor. 

143.  Sometimes  the  series  of  numbers  of  which  the  greatest 
common  divisor  is  to  be  found  cannot  be  factored  by  inspection, 
and  a  method  similar  to  the  following  is  employed. 

144.  Example.      Find    the    greatest    common    divisor   of    697 
and  779. 

697)779(1 

(597  SOLUTION.    Divide  the  greater  number  by  the  less, 

~82^697(8  *^e  Divisor  bv  tne  remainder,  and  so  continue  until 

'        ^  there  is  no  remainder.     The  last  divisor  is  the  greatest 

-— —  common  divisor.     Therefore  the  greatest  common  divi- 

41)82(2  sor  of  697  and  779  is  41. 
82 

WRITTEN  EXERCISE 

Find  the  greatest  common  divisor  of: 

1.  22,  55,  and  99.  8.  679  and  1869. 

2.  24,  36,  60,  and  96.  4-  32,  48,  80,  112,  and  144. 


54  SIMPLE  NUMBERS  [§§  144-147 

5.  A  farmer  has  a  triangular  piece  of  land  which  he  wishes  to 
inclose  with  a  board  fence  so  that  the  boards  may  be  of  the  greatest 
length  possible  and  no  fractional  lengths  used.     If  the  sides  of  the 
tract  of  land  are  84,  96,  and  108  feet,  respectively,  what  is  the  length 
of  the  longest  board  that  can  be  used  in  making  the  inclosure  ? 

6.  How  many  boards  will  inclose,  without  waste,  a  rectangular 
garden,  98  feet  long  by  70  feet  wide,  the  fence  being  straight  and  5 
boards  high,  if  the  boards  be  of  equal  length  and  the  longest  possible  ? 


LEAST  COMMON  MULTIPLE 

145.  General  Principles.     1.  The  product  of  two  or  more  numbers, 
or  any  number  of  times  their  product,  is  a  common  multiple  of  those 
numbers. 

2.  Two  or  more  numbers  may  have   any  number  of  common 
multiples  but  only  one  least  common  multiple. 

3.  A  multiple  of  a  number  contains  all  the  prime  factors  of  that 
number. 

4.  A  common  multiple  of  two  or  more  numbers  contains  all  the 
prime  factors  of  each  of  the  numbers. 

5.  The  least  common  multiple  of  two  or  more  numbers  is  the 
least  number  that  will  contain  all  the  prime  factors  of  the  given 
numbers. 

146.  To  find  the  least  common  multiple  of  two  or  more  numbers. 

147.  Example.     Find  the  least  common  multiple  of  12,  16,  63, 

and  90. 

/a\  SOLUTIONS,    (a)  Since  no  number  less 

12  —  2       2       3  than  ^°  can  be  divic*ed  by  90,  it  is  evident 

that  the  least  common  multiple  cannot  be 
less  than  that  number;    hence,   it    must 

63  =  3  X  3  X  7  contain  the  prime  factors  3,  3,  2,  and  5  in 

90  =  3x3x2x5  order  to  be  divisible  by  90 ;  it  must  con- 

90x7x2x2x2  =  5040     te*n  7  as  a  factor  in  order  to  be  divisible 

by  63 ;  and  it  must  contain  2  as  a  factor 

three  more  times  in  order  to  be  divisible  by  16  ;  hence,  the  product  of  the  factors 
3,  3,  2,  5,  7,  2,  2,  and  2,  or  6040,  must  be  the  least  common  multiple  of  the  num- 
bers 12,  16,  63,  and  90.  Or, 


§§  147-148]  PROPERTIES   OF   NUMBERS  55 

(fo  (6)  First  divide  by  2  ;  63  not  being  divisible  by 


.jo  _  IQ  _  gQ  _  on 


8—63—45 


2,  bring  it  to  the  lower  line  and  divide  again  by 
2  ;  neither  63  nor  45  being  divisible  by  2,  bring  both 
to  the  quotient  line.  Next  divide  by  3  ;  4  not  being 
divisible  by  3,  bring  it  to  the  quotient  line  and 
1  —  4  —  21  —  15  divide  again  by  3.  The  remaining  numbers,  4,  7, 


3—  4—63—45 


1 —   4 —   J —  5     and  5,  being  relatively  prime,  should   be  taken 
together  with  the  prime  divisors  2,  2,  3,  and  3  as 

factors  of  the  least  common  multiple.     Their  product  is  6040,  the  same  as  found 
in  147. 

NOTE.     The  latter  method  of  determining  the  least  common  multiple  will, 
in  the  majority  of  cases,  be  found  the  most  convenient  for  practical  purposes. 
When  one  of  the  given  numbers  is  a  factor  of  another,  reject  the  smaller  one. 

148.    Therefore  the  following  rule  : 

Write  the  numbers  in  a  horizontal  line,  separating  them 
by  dashes. 

Divide  by  any  factor  common  to  all  the  numbers,  or  by 
any  prime  factor  common  to  two  or  more  of  them/. 

In  the  same  manner  divide  the  quotients  obtained,  and 
continue  the  process  until  the  quotients  are  relatively  prime. 

The  product  of  the  several  divisors  and  the  undivided 
numbers  is  the  least  common  multiple. 

WRITTEN  EXERCISE 

Find  the  least  common  multiple  of : 

1.  12,  20,  and  32. 

2.  25,  90,  and  225. 

3.  6,  16,  and  26. 

4.  A,  B,  and  C  are  traveling  men.     A  makes  a  visit  to  Boston 
every  four  months,  B  every  three  months,  and  C  every  two  months, 
tf  they  are  all  in  Boston  on  January  1,  1903,  when  will  they  all  be 
in  that  city  together  again  for  the  first  time  ? 

5.  What  is  the  least  number  of  acres  that  a  piece  of  land  can 
contain  to  be  exactly  divided  into  lots  of  12, 14,  and  18  acres  respec- 
tively ? 


56  SIMPLE   NUMBERS  [§§  149-152 

CANCELLATION 

149.  Cancellation  is  the  process  of  shortening  the  operation  of 
division,  or  the  combined  operations  of  multiplication  and  division, 
by  omitting  or  striking  out  equal  factors  from  the  dividend  and 
divisor. 

150.  General  Principles.      1.   Canceling  a  factor  from  a  number 
has  the  effect  of  dividing  the  number  by  that  factor. 

2.   Canceling  a  factor  from  both  dividend  and  divisor  does  not 
affect  the  value  of  the  quotient. 

151.  Example.     Divide  18  x  36  by  3  x  32. 

(6)  SOLUTION.     Indicate  the  pro- 

9  cess   by  either  of  the   methods 

jo  .     shown  in  the  margin. 

3  and  36  contain  the  common 


factor  3,  which  cancel  and  write 
12  in  the  dividend.  12  and  32 
contain  the  common  factor  4, 
27  =  6j  which  cancel  from  both  and  write 
the  factors  3  and  8  in  the  divi- 
dend and  divisor,  respectively.  18  and  8  contain  the  common  factor  2,  which 
cancel  from  both  and  write  the  factors  9  and  4  in  the  dividend  and  divisor, 
respectively.  The  product  of  the  remaining  factors  of  the  dividend,  divided 
by  the  remaining  factors  of  the  divisor,  is  the  required  quotient,  or  6f. 

152.   Hence  the  following  rule  : 

Indicate  the  operation  in  convenient  form. 

Cancel  from  the  dividend  and  divisor  all  factors  common 
to  both. 

Divide  the  product  of  the  remaining  factors  of  the  divi- 
dend by  the  product  of  the  remaining  factors  of  the  divisor. 

The  result  obtained  is  the  required  quotient. 

WRITTEN  EXERCISE 

1.  Divide  the  product  of  9,  8,  12,  and  24  by  the  product  of  2, 
14,  and  8. 

2.  What  is  the  quotient  of  35  X  75  -5-  7  X  5  x  3  ? 


§152]  PROPERTIED   I'F    NUMBERS  57 

S.  Multiply  together  18,  42,  and  64,  and  divide  the  product  by 
the  product  of  6,  16,  and  32. 

12x3x35x24  x  2=  ? 
6x9x5x6 

5.  How  many  bushels  of  potatoes  at  60^  per  bushel  will  pay 
for  450  pounds  of  sugar  at  6^  per  pound  ? 

6.  A  farmer  traded  4  hogs  weighing  325  pounds  at  6^  per 
pound  for  sugar  at  5  ^  per  pound.     How  many  entire  barrels  of  312 
pounds  each  should  the  farmer  have  received  ? 

7.  How  many  yards  of  cloth  at  15^  per  yard  should  be  given 
for  9  barrels  of  pork,  each  barrel  containing  200  pounds,  at  6^  per 
pound? 

8.  How  many  pieces  of  cloth  containing  45  yards  each  should 
be  received  for  5  baskets  of  eggs,  each  basket  containing  21  dozen 
at  18^  per  dozen,  if  the  cloth  be  valued  at  8^  per  yard  ? 

9.  If  320  acres  of  land  produce  25,600  bushels  of  wheat,  how 
many  bushels  of  wheat  will  110  acres  produce  at  the  same  rate  ? 

10.  If  a  horse  trots  5  miles  in  30  minutes,  how  far  can  he  trot  in 
21  minutes  at  the  same  rate  ? 

11.  How  many  sections  of  Texas  prairie  land,  each  containing 
640  acres,  at  $  5  per  acre,  should  be  given  for  an  Ohio  farm  of  400 
acres,  at  $  40  per  acre  ? 

12.  A  farmer  exchanged  196  loads  of  oats,  each  load  containing 
30  sacks  of  2  bushels  each,  worth  30^  per  bushel,  for  flour  at  5^  per 
pound.     At  196  pounds  per  barrel,  how  many  barrels  should  he  have 
received  ? 

18.  A  farmer  exchanged  250  bushels  of  wheat  at  80  ^  per  bushel 
for  cloth  at  40^  per  yard.  How  many  yards  of  cloth  should  he 
have  received? 

14.  If  9  men  earn  $  108  in  6  days,  how  much  will  15  men  earn 
in  4  days  at  the  same  rate  ? 

15.  30  half  chests  of  Japan  tea  containing  75  pounds  each  at  30^ 
per  pound  were  exchanged  for  brown  sugar  at  2\$  per  pound.     If  a 
barrel  of  brown  sugar  weighs  300  pounds,  how  many  barrels  should 
have  been  received  ? 


UNITED   STATES   MOINEY 

153.  Money  is  a  standard  measure  of  value  used  as  a  medium  of 
exchange. 

154.  Currency  is  the  term  applied  to  money  or  its  equivalent. 

155.  A   decimal  currency   is    a   currency   whose    denominations 
increase  and  decrease  on  a  scale  of  ten. 

156.  United  States  money,  commonly  called  Federal  money,  is  the 
legal  currency  of  the  United  States.     It  is  a  decimal  currency  and 
consists  of  coin  and  paper  money. 

157.  The  denominations  and  scale  of  United  States  money  are 

shown  in  the  following 

TABLE 

10  mills  (m.)  =1  cent  (j>  or  ct.) 

10  cents  =  1  dime  (d.) 

10  dimes  or  100  cents  =  1  dollar  ($) 
10  dollars  =  1  eagle  (E.) 

The  dollar  sign,  $,  is  always  written  before  the  number.  The  mill  is  not  a 
coin.  It  is  used  only  as  a  decimal  of  a  cent,  which  is  the  smallest  money  of 
the  mint  and  the  smallest  recognized  in  business.  The  eagle  and  the  dime  are 
used  only  as  names  of  coins  and  never  in  reading  United  States  money. 

158.  Bullion  is  pure  gold  or  silver  in  bars  or  ingots. 

159.  Coins  are  pieces  of  metal  converted  into  money  by  being 
stamped  by  the  authority  of  the  government  in  such  a  way  as  to 
indicate  the  rate  at  which  they  shall  pass  in  trade. 

160.  The  coins  of  the  United  States  are  of  two  kinds,  namely : 

1.  Those  made  by  the  authority  of  the  government,  in  unlimited 
quantities,  for  private  persons  from  metal  deposited  by  them. 

The  government  provides  that  private  persons  may  deposit  metal  with  the 
United  States  mints  or  assay  offices  in  unlimited  quantities  for  the  purpose  of 
having  it  weighed,  refined,  assayed,  and  returned  to  them  in  the  form  of  stand- 
ard coins  or  in  ingots  of  standard  fineness. 

2.  Those  subsidiary  coins  made  from  silver,  nickel,  and  copper 
by  the  authority  of  the  government. 

58 


§§  161-167] 


UNITED  STATES  MONEY 


59 


The  government  has  the  power  to  buy  metal  for  the  purpose  of  making  it 
into  subsidiary  coins  for  itself.  Since  subsidiary  coins  may  be  sold  to  private 
individuals  at  more  than  cost,  their  quantity  is  restricted. 

161.   The  coins  of  the  United  States  are  gold,  silver,  bronze,  and 

nickel. 

COINS  OF  THE  UNITED  STATES 


COINS 

COMPOSITION 

WEIGHT 

VALUE 

Gold 

Quarter-eagle 

&  pure  gold  and  ^  alloy 

64.5    Troy  grains 

$2.50 

Half-eagle 

^  pure  gold  and  ^  alloy 

129       Troy  grains 

6.00 

Eagle 

T%  pure  gold  and  ^  alloy 

258       Troy  grains 

10.00 

Double-eagle 

^j  pure  gold  and  ^  alloy 

516       Troy  grains 

20.00 

Silver 

Dime 

r?5  pure  silver  and  ^  alloy 

38.58  Troy  grains 

$0.10 

Quarter-dollar 

T%  pure  silver  and  ^  alloy 

96.45  Troy  grains 

.25 

Half-dollar 

T9^  pure  silver  and  ^  alloy 

192.9    Troy  grains 

.50 

Dollar 

^  pure  silver  and  -J^  alloy 

412.5    Troy  grains 

1.00 

Bronze  and  Nickel 

1-cont  piece 

^  copper  and  -fa  tin  and  zinc 

48       Troy  grains 

$0.01 

5-cent  piece 

|  copper  and  \  nickel 

77.16  Troy  grains 

.05 

162.  The  standard  unit  of  value  in  the  United  States  is  the  gold 
dollar,  which  contains  23.22  Troy  grains  of  pure  gold  and  weighs 
25.8  Troy  grains. 

The  gold  dollar  being  so  inconveniently  small  is  not  coined  now. 

163.  The  paper  money  of  the  United  States  at  present  consists 
of  gold  certificates,  silver  certificates.  United  States  notes,  treasury  notes, 
and  national  bank  bills. 

164.  Gold    certificates   are   issued   for  gold  deposited  with  the 
Treasurer  of  the  United  States.     They  represent  values  of  $20  and 
up  ward  to  $20,000. 

165.  Silver  certificates  are  issued  for  silver  deposited  with  the 
Treasurer  of  the  United  States  in  amounts  not  less  than  $10.     They 
represent  values  of  $  1  and  upward  to  $  100. 

166.  United  States  notes  (greenbacks)  represent  values  of  $10 
and  upward  to  $1000. 

167.  National  bank  bills  are  notes  issued  by  national  banks  under 
the  supervision  of  the  national  government.     They  are  now  issued 
in  amounts  of  $5  and  upward  to  $100. 


60  UNITED   STATES    MONEY  [§§  1C8-172 

168.  Treasury  notes  of  1890  are  now  in  the  course  of  retirement. 
They  were  formerly  issued  in  amounts  of  $  1  and  upward  to  $20. 
They  cannot  be  reissued. 

169.  Legal  tender  is  the  term  applied  to  such  money  as  may  be 
legally  offered  in  payment  of  debts. 

170.  The  following  are  legal  tender  in  the  United  States  as  noted : 

1.  Gold  coins,  except  when  below  the  standard  weight  because 
of  abrasion. 

2.  Silver  dollars  and  treasury  notes  of  1890  in  all  cases  where 
the  contract  does  not  expressly  stipulate  otherwise. 

3.  United  States  notes  (greenbacks),  except  for  interest  on  the 
public  debts  and  for  duties  on  imports. 

4.  National  bank  notes  for  any  debt  to  a  national  bank,  and  for 
taxes  and  other  dues  to  the  United  States,  except  duties  on  imports. 

5.  Silver  coins  less  than  one  dollar  in  all  cases  where  the  amount 
does  not  exceed  ten  dollars  in  one  payment. 

6.  Nickel  and  copper  coins  in  all  cases  where  the  amount  does 
not  exceed  twenty-five  cents  in  one  payment. 

NOTATION  OP  UNITED  STATES  MONEY 

171.  The  dollars  form  the  integral  part  of  the  number,  and  are 
written  to  the  left  of  the  dot,  called  the  decimal  point.     The  cents 
and  mills  form  the  fractional  part  of  the  number  and  are  written  to 
the  right  of  the  decimal  point. 

172.  Since  ten  dimes  make  one  dollar,  the  figures  written  in  the 
first  place  to  the  right  of  the  decimal  point  are  tenths  of  a  dollar,  or 
dimes.     Since  one  hundred  cents  make  one  dollar,  the  figures  written 
in  the  second  place  to  the  right  of  the  decimal  point  are  hundredths  of  a 
dollar,  or  cents.     Since  one  thomand  mills  make  a  dollar,  the  figures 
written  in  the  third  place  to  the  right  of  the  decimal  point  are  thou- 
sandths of  a  dollar,  or  mills. 

Thus:  Six  dollars,  forty-eight  cents,  two  mills,  is  written  $6.482. 

In  writing  cents  less  than  ten,  a  cipher  should  occupy  the  first  place  to  the 
right  of  the  decimal  point. 

In  the  final  results  all  mills  less  than  five  are  rejected,  and  all  five  or  more 
are  counted  as  a  whole  cent. 


§§  173-178]  UNITED   STATES    MONEY  61 

173.  Whenever  it  is  desirable  to  express  United  States  money 
in  written  words,  the  cents  should  be  written  as  hundredths  of  a 
dollar,  and  in  fractional  form. 

Thus :  Twenty-live  and  j$$  dollars. 

REDUCTION  OF  UNITED  STATES  MONEY 

174.  Reduction   is   the   process   of   changing   the    unit  without 
changing  the  value  of  a  number. 

175.  To  reduce  dollars  to  cents. 

176.  Since  there  are  one  hundred  cents  in  a  dollar,  to  reduce 
dollars  to  cents,  multiply  by  100  by  annexing  two  ciphers  to  the  numbers 
expressing  a  whole  number  of  dollars,  or  by  moving  the  decimal  point 
two  places  to  the  right  in  numbers  expressing  cents,  or  dollars  and  cents. 

177.  To  reduce  cents  to  dollars, 

Divide  by  100  by  removing  the  decimal  point  two  places  to  the  left. 

ORAL  EXERCISE 
By  inspection,  reduce : 

1.   $7.25  to  cents.         3.    241 0  to  dollars.          5.   $  157.32  to  cents. 
%.   $119  to  cents.         4.    92,798^  to  dollars.     6.   72,572^  to  dollars. 

7.  Which  is  the  heavier,  a  gold  eagle  or  a  silver  dollar  ?     A 
gold  eagle  or  a  silver  half-dollar  ? 

8.  Which  is  the  heavier,  a  five-dollar  gold  piece  or  a  five-cent 
nickel  piece  ?  a  gold  dollar  or  a  bronze  cent  ? 

9.  Why  are  the  gold  and  silver  coins  of  the  United  States  never 
made  more  than  T%  pure  ? 

10.  Are  national    bank    notes    legal    tender  ?    silver  dollars  ? 
pennies  ?   gold  certificates  ?   silver  certificates  ? 

11.  Under  what  circumstances  would  gold  coin  not  have  full 
legal  tender  value? 

ADDITION  AND  SUBTRACTION  OF  UNITED  STATES  MONEY 

178.  To  add  or  subtract  United  States  money, 
Write  dollars  under  dollars  and  cents  under  cents. 

Add  or  subtract  as  in  simple  numbers,  placing  the  decimal  point  in 
the  result  directly  under  the  points  in  the  numbers  added  or  subtracted. 


62 


UNITED  STATES  MONEY 


[§178 


*, 

S. 

4- 

$47;198.76 

$919,010.01 

$876,311.40 

2.91 

1,889.76 

40.32 

1,487.59 

33.44 

21.56 

101.11 

221.34 

2,197.77 

321,876.34 

2,345.66 

140.40 

8,198.99 

1.06 

999.88 

2,345.98 

66.87 

278,811.33 

1.11 

221,198.32 

9,435.23 

231.59 

44,859.83 

1.06 

22.81 

321.54 

6.05 

45,728.67 

2,378.95 

200,400.58 

3,221.55 

34.40 

3,211.59 

19.88 

70.87 

20.87 

987,111.23 

1,100.58 

12.549.15 

WRITTEN  EXERCISE 

Copy  or  write  from  dictation  and  find  the  sum  of  each  of  the 
following  problems :  * 

1. 

$157,926.04 
52.19 

9,261,549.62 

1,694,247.57 

5,216.90 

425.86 

52.95 

1,076.87 

27,214.95 

276,421.87 

932.17 

4.26 

259,426.74 
9,275.18 

Copy  or  write  from  dictation  the  following  numbers  and  find  the 
sums  by  horizontal  addition :  f 

5.  93210,  22,11s03,  8121,  967s2,  221» 

6.  34650,  29175,  10031,  26911,  8093. 

7.  216584,  7243,  9020,  11765,  600"    1127". 

8.  1516,  2s7,  II46,  10790,  9*,  81",  123"2,  601,  1580,  lln. 

9.  A  merchant  bought  cottons,  for  3467s25 ;  linens,  for  132615; 
woolens,  for  421575;  delaines,  for  102548;  brocades,  for  112750.  If  all 
were  sold  for  1325628,  how  much  was  gained  ? 

*  Addition  is  so  interwoven  with  all  arithmetical  processes  that  proficiency  in 
the  subject  should  be  insisted  upon.  If  the  principles  of  grouping  have  not  been 
mastered,  simple  addition  should  be  reviewed  before  any  advance  steps  are  taken. 
Accurate  answers  for  problems  similar  to  1,  2,  3,  and  4  should  be  obtained  in  from 
twenty  to  twenty-five  seconds. 

t  Under  some  circumstances  it  is  desirable  to  write  United  States  money  expressed 
Jn  dollars  and  cents  without  the  dollar  sign  and  the  decimal  point,  with  the  decimal 
part  placed  slightly  above  that  expressing  the  integers  or  dollars.  • 

Thus,  $5.25  may  be  written  S26.  $13.08  may  be  written  13«8.  This  is  advisable 
only  where  the  sum  of  several  items  is  to  be  found  by  horizontal  addition. 


§  179]  UNITED   STATES   MONEY  t>3 

MULTIPLICATION  OF  UNITED  STATES  MONEY 

179.   To  multiply  United  States  money, 

Multiply  as  in  simple  numbers,  and  from  the  right  in  the  product 
point  off  as  many  places  as  there  are  places  to  the  right  of  the  decimal 
point  in  the  multiplicand. 

Money  is  a  concrete  expression.  In  a  critical  analysis  of  its  multipli- 
cation, therefore,  the  money  cost  or  price  of  the  article  is  a  concrete  multiplicand. 
The  number  of  things  bought  or  sold  is  an  abstract  multiplier  and  their  product 
is  concrete  and  of  the  same  name  or  denomination  as  the  multiplicand.  However, 
since  the  United  States  money  scale  is  decimal,  these  terms  may  be  interchanged 
for  convenience. 

WRITTEN  EXERCISE 

1.  Find  the  amount  of  the  following  bill.    Make  all  extensions 
mentally  as  explained  in  85-88. 

28  Ib.  lard  at  11  £  112  Ib.  butter  at  22  £ 

46  bu.  salt  at  22^.  132  bu.  onions  at  44^. 

117  bu.  apples  at  33^.  113  bu.  potatoes  at  66^. 

2.  Find  the  total  cost  of  the  following  farm  produce.     Make  all 
extensions  mentally  as  explained  in  95-97. 

24  bu.  wheat  at  $  1.02.  103  bu.  rye  at  83  £ 

215  bu.  barley  at  $1.04.  105  bu.  peas  at  72^. 

108  bu.  oats  at  42^.  204  Ib.  butter  at  34  £ 

3.  Find  the  amount  of  the  following  bill.     Make  all  extensions 
mentally  as  explained  in  91-93. 

54  yd.  jeans  at  21  £  48  yd.  print  at  24^. 

27  yd.  delaine  at  32  £  121  yd.  ticking  at  15  tf. 
64  yd.  gingham  at  23  £  61  yd.  sheeting  at  13  £ 

42  yd.  drilling  at  21  f.  217  yd.  cashmere  at  $1.13. 

4-  Find  the  total  cost  of  the  following  items.  Make  the  exten- 
sions mentally  as  explained  in  95-97. 

116  yd.  moquette  carpet  at  $  3.02.  115  yd.  border  No.  1  at  $  3.06. 
131  yd.  Brussels  carpet  at  $  2.05.  64  yd.  border  No.  2  at  $2.07. 
103  yd.  ingrain  carpet  at  $1.04.  43  yd.  border  No.  3  at  $1.08. 

5.   Find  the  amount  of  the  following  bill.     Make  all  extensions 
X  as  explained  in  90. 

325  yd.  tapestry  Brussels  at  $  2.17.   648  yd.  Axminster  at  $  2.55. 
547  yd.  3-ply  ingrain  at  $  1.26.         328  yd.  body  Brussels  at  $  2.48. 
427  yd.  moquette  at  $  2.79.  255  yd.  velvet  at  $  2.79. 


64  UNITED   STATES  MONEY  [§§  180-182 

DIVISION  OF  UNITED  STATES  MONEY 

180.  To  divide  United  States  money. 

181.  Examples.     1.  If  5  hats  are  worth  $25.65,  what  is  1  hat 

worth  ? 

g  13  SOLUTION.    Since  5  hats  are  worth  2565  cents  ($25.65),  1 

—  -^—  -        hat  is  worth  £  of  2565  cents,  or  613  cents.     613  cents  equals 


2.   If  4  boxes  of  oranges  are  worth  $17,  what  is  1  box  worth? 
4  OK  SOLUTION.      Since  4  boxes  of    oranges    cost    1700    cents 

($17.00),  one  box  will   cost  |  of  1700   cents,  or  425  cents. 


A\1  7  nrt  ., 

425  cents  equals  $4.25. 

8.   How  many  boxes  of  oranges  at  $  4.25  per  box  can  be  bought 

for  $17? 

.  SOLUTION.    Since  1  box  of  oranges  cost  425  cents  ($4.25), 

-     as  many  times  1  box  can  be  bought  for  1700  cents  as  425  is  con- 
tained times  in  1700,  or  4  times. 

182.   Therefore  the  following  rule  : 

Divide  as  in  simple  numbers. 

From  the  right  of  the  quotient  point  off  as  many  decimal 
places  as  the  number  of  places  in  the  dividend  exceed  those 
in  the  divisor. 

When  the  dividend  and  divisor  each  contain  the  same  number  of  decimal 
places,  the  numbers  may  be  regarded  as  integers  in  performing  the  operation  of 
division. 

When  the  divisor  alone  contains  cents,  both  dividend  and  divisor  may  be 
reduced  to  cents  and  the  division  performed  exactly  as  in  whole  numbers. 

WRITTEN  EXERCISE 

1.  A  dealer  bought  wheat  at  95^,  oats  at  45^,  and  corn  at  65^ 
per  bushel.     He  paid  $332.50  for  the  wheat,  $191.25  for  the  oats, 
and  $  113.75  for  the  corn.     How  many  bushels  of  each  did  he  buy 
in  all? 

2.  Having  sold  my  mill  for  $17,250,  and  316  barrels  of  flour  in 
stock  at  $5.15  per  barrel,  I  invested  out  of  the  proceeds  $  1185.85  in 
furnishing  a  house,  $  1260  in  farming  utensils,  $  1582.25  in  live  stock, 
and  with  the  remainder  paid  in  full  for  a  farm  of  163  acres.     What 
was  the  cost  of  the  farm  per  acre  ? 


METHODS  FOR  PROVING  WORK 

183.  Addition  is  generally  verified  as  explained  in  44. 

184.  A  good  way  to  test  the  accuracy  of  subtraction  is  to  add 
the  remainder  and  the  subtrahend.    If  the  work  is  correct,  the  result 
obtained  should  equal  the  minuend. 

185.  The  work  of  multiplication  may  be  verified  by  interchang- 
ing the  multiplier  and  the  multiplicand,  and  remultiplying.     If  the 
results  obtained  by  both  operations  are  the  same,  the  work  is  assumed 
to  be  correct. 

186.  The  operation  of  division  may  be  verified  by  multiplying 
together  the  quotient  and  divisor  and  adding  to  the  product  the 
remainder,  if  any. 

CASTING  OUT  NINES  AND  ELEVENS 

187.  The  basis  of  our  numerical  system  being  10,  every  power  * 
of  10  is  1  more  than  some  multiple  of  9 ;  and  10  or  any  power  of  10 
multiplied  by  a  single  digit  is  some  multiple  of  9,  plus  that  digit. 

Thus,  10  =  9  +  1 ;   100  =  11  x  9  +  1 ;  and  60  =  6  x  9  +  6  ;  500  =  55  x  9  -f  5. 

188.  4582  =  4000  +  500  +  80  +  2.     Since  the   excess   of  nines 
(the  remainder  after  the  nines  are  cast  out)  in  4000  is  4,  in  500, 
5,  in  80,  8,  in  2,  2,  it  follows  that  the  excess  of  nines  in  any  num- 
ber is  the  same  as  the  excess  of  nines  in  the  sum  of  the  digits  of 
that  number. 

Thus,  the  excess  of  nines  in  4682  =  4  +  5  +  8  -f-  2,  or  19.     19  =  10.     10=1. 

189.  In  finding  the  excess  of  nines,  it  is  always  best  to  omit  all 
nines  and  also  to  reject  them  as  soon  as  they  occur  in  the  addition. 

Thus,  in  casting  out  the  nines  in  954,727,  begin  at  the  left  and  drop  the  nines 
as  soon  as  they  occur.    The  excess  of  nines  in  954,727  is  then  found  to  be  7. 

*  A  power  of  a  number  is  the  product  arising  from  multiplying  a  number  by 
itself  one  or  more  times. 


66  METHODS  FOR  PROVING  WORK  [§§  190-197 

190.  Since  11  is  just  1  more  than  the  numerical  basis  10,  even 
powers  of  10  are  multiples  of  11,  plus  1,  and  odd  powers  of  10  are 
multiples  of  11,  minus  1. 

Thus,  10  x  10  or  100,  an  even  power  of  10,  equals  11x9+1.  10  x  10  x  10 
X  10  x  10,  an  odd  power  of  10,  equals  9091  x  11  -  1. 

191.  Any  even  power  of  10  multiplied  by  a  single  digit  is  some 
multiple  of  11  plus  that  digit.     Any  odd  power  of  10  multiplied  by 
a  single  digit  is  some  multiple  of  11  minus  that  digit. 

Thus,  600  =  54  x  11  +  6 ;  and  6000  =  546  x  11  -  6. 

192.  10  multiplied  by  any  single  digit  is  some  multiple  of  11 
minus  the  difference  between  11  and  that  digit. 

Thus,  30  =  2x114-8;  50  =  4x11+6;  80  =  7x11+3;  90  =  8x11+2;  etc. 

193.  It  therefore  follows  that : 

The  digit  in  the  odd  place  of  any  number  of  two  figures,  with  eleven 
added  whenever  necessary,  minus  the  digit  in  the  even  place,  equals  the 
excess  of  elevens  in  that  number. 

Applying  this  principle  to  all  numbers,  the  sum  of  the  digits  in  the 
odd  places,  increased  by  eleven  or  a  multiple  of  eleven  whenever  neces- 
sary, minus  the  sum  of  the  digits  in  the  even  places,  is  equal  to  the  excess 
of  elevens  in  the  entire  number. 

194.  The  elevens  may  be  dropped  from  the  partial  additions  in 
the  same  manner  as  explained  for  nines. 

195.  The  properties  of  nine  and  eleven,  as  explained  above,  may 
be  used  in  proving  addition,  subtraction,  multiplication,  and  division. 

196.  To  prove  addition  by  casting  out  the  nines  and  elevens. 

197.  Example.    Add  375,  425,  623,  and  412.    Prove  the  work  by 
casting  out  (a)  the  nines  and  (b)  the  elevens. 

(a)  Excess  of  nines. 


375=   6 
425=    2 


412=    7 


SOLUTION.    The  excess  of  nines  in  375  Is  6  ;  In  425 
is  2  ;  in  623  is  2  ;  in  412  is  7.    The  excess  of  nines  in 


p     v  the  sum  of  6,  2,  2,  and  7  is  o.     I  he  excess  ( 

1835  is  also  8.     Since  the  excess  of  nines  in  all  the  num- 


bers  is  equal  to  the  excess  of  nines  in  the  sum  of  the 


1835  =  17  =  8  numbers,  the  work  is  assumed  to  be  correct. 


197-201]  METHODS  FOR   PROVING   WORK  67 

(b)  Excess  of  elevens. 


375  s=  1 
425  =  7 
62S  -•  7 


SOLUTION.      16  (11  +  5)  -  7  +  3  =  12.      12  =  1,  or 

tne  excess  of  elevens  in  376.    5  —  2  +  4  =  7,  or  the 
excess  of  elevens  in  425.    3  —  2  +  6  =  7,  or  the  excess 


of  elevens  in  623.     2  —  1  -f-  4  =  5,   or  the  excess  of 
_£  elevens  in  412.    The  sum  of  1,  7,  7,  and  5  equals  20, 

1835  =  20  =  9  which,  divided  by  11,  gives  a  remainder  9.     (5-3) 

•f  (8  —  1)  =  9,  the  excess  of  elevens  in  the  sum.    Since 

the  excess  of  elevens  in  all  the  numbers  is  equal  to  the  excess  of  elevens  in 
the  sum,  the  work  is  assumed  to  be  correct. 

198.  To  prove  subtraction  by  casting  out  the  nines  and  elevens. 

199.  The  excess  of  nines  or  elevens  in  the  minuend  minus  the 
excess  of  nines  or  elevens  in  the  subtrahend  should  equal  the  excess 
of  nines  or  elevens  in  the  remainder;  or,  the  excess  of  nines  or 
elevens  in  the  subtrahend  plus  the  excess  of  nines  or  elevens  in  the 
remainder  should  equal  the  excess  of  nines  or  elevens  in  the  minuend. 

200.  To  prove  multiplication  by  casting  out  the  nines  and  elevens. 

201.  Example.    Find  the  product  of  512  x  324  and  verify  the 
result  by  casting  out  (a)  the  nines  and  (b)  the  elevens. 


(a)  Excess  of  nines. 

512  =  81  SOLUTION.    Use  the  contracted  method  of  multi- 

324  —  0  I  Paying  explained  in  90.     The  excess  of  nines  in  512 

_  =  is  8  ;  in  324,  0.    8x0  =  0.    The  excess  of  nines  in 

2048  the  completed  product  is  0.    Since  the  excess  of  nines 

16384  to  *he  multiplicand  multiplied  by  the  excess  of  nines  in 

the  multiplier  is  equal  to  the  excess  of  nines  in  the 


product,  the  work  is  assumed  to  be  correct. 

Excess  of  elevens. 

_  g  i  SOLUTION.    2  —  1  -f  5  =  6,  or  the  excess  of  elevens 

_  £  f  30  =  8      in  512.     4-2  +  3  =  5,  or  the  excess  of  elevens  in  324. 


(&)   Excess  of  elevens. 

512  =  6  1  SOLUTION.    2  —  1  -f  5  =  6,  or  the  excess  of  elevens 

324  — 

-  •  -  6  x  6  =  30,  which,  divided  by  11,  gives  a  remainder  8. 

8-8  +  8  —  5-1-6  —  1  =  8,  or  the  excess  of  elevens  in 
the  completed  product.  Since  the  excess  of  elevens 

"165888  =  8  in  the  multiplicand  multiplied  by  the  excess  of  elevens 

in  the  multiplier  is  equal  to  the  excess  of  elevens  in 

the  completed  product,  the  work  is  assumed  to  be  correct. 


68  METHODS  FOR   PROVING    WORK  [§§202-203 

202.  To  prove  division  by  casting  out  the  nines  and  elevens. 

203.  Examples  ID  division  may  be  proved  by  multiplying  the 
excess  of  nines  or  elevens  in  the  divisor  by  the  excess  of  nines 
or  elevens  in  the  quotient.     If  the  work  is  correct,  the  result  should 
equal  the  excess  of  nines  or  elevens  in  the  dividend,  or  the  dividend 
minus  the  remainder  when  there  is  a  remainder. 

On  the  whole,  the  proof  of  casting  out  the  elevens  is  more  reliable  than  tbe 
proof  of  casting  out  tbe  nines,  but  neither  proof  is  practiced  very  generally  by 
accountants  except  in  proving  long  multiplications  and  divisions. 

WRITTEN  EXERCISE 

1.  Multiply  125,426  by  567  in  two  lines  of  partial  products. 
Verify  the  work  by  casting  out  the  elevens. 

2.  Multiply  112,121  by  12,816  in  two  lines  of  partial  products. 
Verify  the  work  by  casting  out  the  nines. 

8.  Multiply  121,214  by  112,568  in  three  lines  of  partial  products. 
Verify  the  work  by  casting  out  the  elevens. 

4.  Find  the  amount  of  the  following  bill,  making  the  extensions 
mentally  as  explained  in  90.     Verify  the  work  by  casting  out  the 
elevens. 

248  yd.  black  dress  silk,  $1.24.     248  yd.  black  wool  crepon,  $2.79. 
576  yd.  Amazon  cloth,  $  2.17.        568  yd.  English  camel's  hair,  $  2.17. 
357  yd.  cashmere,  $1.55.  124  doz.  cotton  hose,  $1.86. 

5.  Find  the  amount  of  the  following  bill,  making  the  extensions 
mentally  as  explained  in  85-88.     Verify  the  work  by  casting  out  the 
nines. 

116  gr.  bone  buttons,  22^.  141  yd.  feather  ticking,  11  £ 

112  pc.  black  Chantilly  lace,  88  £  118  yd.  fancy  gingham,  11  £ 

53  yd.  cotton  surah  lining,  66 £.  54  yd.  gunner's  duck,  22^. 

124  yd.  wash  silk,  44^.  83  yd.  Scotch  cheviot,  55^. 

WRITTEN  REVIEW 

1.  Find  the  total  of  the  products  called  for  in  the  oral  exercise, 
page  40. 

.  2.  Find  the  total  cost  of  the  items  in  the  oral  review,  pages  45 
and  46. 


§  203]  METHODS  FOR   PROVING   WORK  69 

3.  A  man  earned  $312.50  during  February.     During  March  he 
earned  $49.50  more  than  in  February.     In  April  he  earned  as  much 
as  he  did  during  February  and  May.     If  he  earned  $200  in  May, 
how  much  did  he  earn  in  the  four  months  ? 

4.  A  finds  that  in  five  months  he  spends  as  much  as  he  earns  in 
four.     At  that  rate,  how  long  will  it  take  him  to  save  $  2400  if  he 
earns  $1200  a  year? 

5.  Multiply  12,501  by  486,  making  only  two  lines  of  partial 
products.     Verify  the  work  by  casting  out  the  nines. 

6.  A  man  bought  an  equal  number  of  barrels  of  flour  and  oat- 
meal for  $260.     If  the  flour  cost  $6.20  and  the  oatmeal  $6.80  per 
barrel,  how  many  barrels  of  each  did  he  buy  ? 

7.  A  merchant  failed  in  business,  and  the  excess  of  his  liabilities 
over  resources  was  found  to  be  $3000.    If  he  could  pay  his  creditors 
but  6C  $  on  a  dollar,  what  were  his  total  liabilities,  and  how  much  did 
A,  whom  he  owed  $  1500,  receive  ? 

8.  A  and  B  had  $9245  divided  between  them.     The  difference 
between  their  shares  was  $  245.     What  had  each  ? 

9.  A  merchant's  cash  receipts  for  a  week  were  as  follows :  Mon- 
day, $921.40;  Tuesday,  $525.44;  Wednesday,  $321.50;  Thursday, 
$425.60;    Friday,   $926.80;  Saturday,    $120.40.     What   were   his 
average  daily  sales  ? 

10.  How  many  barrels  of  apples  at  $  2.50  must  be  given  for  1200 
bushels  of  potatoes  at  50  ^  ? 

11.  A  merchant's  gains  for  February  amounted  to  $1600,  or 
$1100  less  than  his  gains  for  January.     If  his  gains  for  January 
were  $60  more  than  four  times  his  gain  for  March,  what  were  his 
total  gains  for  the  three  months  ? 

12.  Multiply  113,214  by  12,816  in  two  lines  of  partial  products. 

18.   Multiply  21,213  by  96,486  in  three  lines  of  partial  products. 
Verify  the  work  by  casting  out  the  elevens. 


FRACTIONS 

COMMON  FRACTIONS 

204.  Quantity  is  anything  which  may  be  measured  or  which  may 
oe  regarded  as  being  made  up  of  parts  like  the  whole.     Numbers  are 
the  expressions  by  which  we  measure  quantities.     The  basis  of  all 
numbers  is  the  unit. 

There  are  integral  units  (3)  or  whole  things,  and  fractional  units 
(5)  or  parts  of  things  obtained  by  dividing  integral  units  into  any 
number  of  equal  parts. 

205.  A  fraction  is  one  or  more  fractional  units. 

206.  A  common  fraction  is  a  fraction  expressed  by  two  numbers, 
one  written  above  and  the  other  below  a  horizontal  line. 

207.  The  terms  of  the  fraction  are  the  two  integers,  called  numer- 
ator and  denominator,  used  to  express  one  or  more  fractional  units. 

208.  The  denominator  is  the  term  which  indicates  the  number  of 
parts  into  which  a  unit  has  been  divided ;  it  is  written  below  the 
line,  and  denominates  or  names  the  size  or  value  of  each  part  of  the 
fraction. 

209.  The  numerator  is  the  term  which  indicates  the  number  of 
equal  parts  taken  to  form  the  fraction;  it  is  written  above  the  line. 

210.  To  read  fractions, 

Pronounce  the  numerator  first  and  then  the  denominator. 
Thus,  } ,  f ,  and  |,  are  read,  one  seventh,  two  thirds,  and  seven  eighths. 

211.  All  fractions  express  unperformed  division.     The  denomi- 
nator is  the  divisor,  the  numerator  the  dividend,  and  the  value  of  the 
number  expressed  by  the  fraction  the  quotient.     Hence, 

When  the  numerator  and  denominator  are  equal,,  the  value 
expressed  by  tlie  fraction  is  1 ;  when  the  numerator  is  less 
than  the  denominator,  the  value  expressed  by  the  fraction  is 

70 


§§211-214]  COMMON  FRACTIONS  71 

less  than  1;  when  the  numerator  is  more  than  the  denomi- 
nator, tlie  value  expressed  by  the  fraction  is  greater  than  1 ; 
and 

Of  two  fractions  Jiaving  the  same  denominator,  the  one 
having  the  larger  numerator  expresses  the  greater  value. 

Of  two  fractions  having  the  same  numerator,  the  one  having 
the  smaller  denominator  expresses  the  greater  value. 

DRILL  EXERCISE 

1.  What  is  the  unit  of  5  tons  of  coal  ?  of  5  thousand  feet  of 
lumber  ?  of  7  dozen  of  eggs  ?  of  -|  of  a  week  ?  of  2^-  acres  of  land  ? 

2.  What  is  the  fractional  unit  of  a  number  divided  into  3  parts  ? 
into  31  parts  ?  into  13  parts  ? 

8.   What  is  the  fractional  unit  of  f  ?  of  -&  ?  of  ^  ?  of  f  ? 

4.  A  unit  contains  how  many  sevenths  ?  fifteenths  ?  elevenths  ? 
twenty-fifths?  eighths?  sixths? 

5.  Which  is  the  greater,  1  of  a  number  or  1  of  a  number  ?    Why  ? 

6.  How  many  times  -fa  of  a  number  is  1  ?     Why  ? 

7.  A  receives  -J-  of  the  profits  of  a  business,  and  B  i.     If  B's 
profits  in  one  month  are  $400,  what  are  A's  profits  for  the  same 
time? 

SOLUTION.  When  a  number  is  divided  into  3  equal  parts,  each  part  is  twice 
as  large  as  the  pa"rts  of  the  same  unit  divided  into  6  equal  parts.  Hence,  |  of  a 
number  is  twice  £  of  the  same  number  ;  and  if  £  of  a  number  is  $400,  ^  of  the 
same  number  is  twice  $400,  or  $800.  Therefore,  A's  profits  are  $800. 

8.  A  invested  -fa  in  the  capital  stock  of  a  certain  business,  and 
B  f     If  A's  investment  was  $  9000,  what  was  B's  ?     Why  ? 

CLASSIFICATION  OF  FRACTIONS 

212.  For  convenience  fractions  may  be  classified  as  simple,  com- 
plex, and  compound. 

213.  A  simple  fraction  is  a  fraction  which  has  one  numerator 
and  one  denominator,  each  of  which  is  an  integer. 

Simple  fractions  are  either  proper  or  improper. 

214.  A  proper  fraction  is  a  fraction  whose  numerator  is  less  than 
its  denominator,  and  whose  value  is  less  than  1 ;  as,  f,  ^,  f . 


72  FRACTIONS  [§§  215-225 

215.  An  improper  fraction  is  a  fraction  whose  numerator  is  equal 
to,  or  greater  than,  its  denominator,  and  whose  value  is  1,  or  more 
than  !;«,¥>¥»¥• 

216.  A  mixed  number  is  a  whole  number  and  a  fraction  united ; 
as,  21  74,  2501 

217.  A  complex  fraction  is  a  fraction  having  one  or  both  of  its 

terms  fractional  j  as,  I,    -,   5,   _X 
9    I    i    A 

218.  A  compound  fraction  is  a  fractional  part  of  a  whole  number 
or  mixed  number,  or  another  fraction ;  as,  |  x  ^,  £  x  2 J,  f  of  T97. 

219.  General  Principles.     1.  Multiplying  the  numerator  or  divid- 
ing the  denominator  multiplies  the  fraction. 

2.  Dividing  the  numerator  or  multiplying  the  denominator  di- 
vides the  fraction. 

3.  Multiplying  or  dividing  both  the  numerator  and  denominator 
by  the  same  number  does  not  alter  the  value  of  the  fraction. 

REDUCTION  OF  FRACTIONS 

220.  The  process  of  changing  the  form  without  changing  the 
value  of  fractions  is  called  reduction  of  fractions. 

221.  To  reduce  a  whole  number  to  a  fraction. 

222.  Example.     Reduce  7  to  a  fraction  whose  denominator  is  7. 

SOLUTION.     Since  in  1  unit  there  are  7  sevenths,  in  7  units  there  must  be 
7  times  7  sevenths,  or  49  sevenths.     Hence,  7  equals  -*£-. 

223.  The  value  of  a  mixed  number  may  be  represented  by  an 
improper  fraction. 

224.  To  reduce  a  mixed  number  to  an  improper  fraction. 

225.  Example.     Keduce  17|  to  halves. 

17^-  SOLUTION.     Since  in  1  there  are  2  halves,  in  17  there  must  be 

35  17  times  2  halves,  or  34  halves.     Thirty-four  halves  plus  1  half 

~  equals  35  halves.     Hence,  17£  equals  ^. 


§§225-228]  COMMON   FRACTIONS  73 

ORAL  EXERCISE 

1.  How  many  fourths  in  6  ?  in  25  ?   in  16  ?    in  71  ?    in  12J  ? 

2.  How  many  thirds  in  15  ?  fourths  ?   fifths  ?   sixths  ?   ninths  ? 
S.   How  many  thirds  in  12J?   in  8£?   in!7£?   in42|?   in27|? 
^.   How  many  sixths  in  2J-  ?  in  1  £  ?    in  4J  ?   in  51  ?   in  12|  ? 

£.  Nine  equals  how  many  times  %  ?  how  many  times  £  ? 

&  How  many  more  pieces  in  5  melons  cut  into  fourths  than  in 
3  melons  cut  into  fifths  ? 

7.  A  has  2J  tons  of  coal  which  he  proposes  to  distribute  among 
some  poor  families.  How  many  families  will  he  aid  if  he  gives  each 
one  £  of  a  ton  ? 

WRITTEN  EXERCISE 

Reduce  the  following  mixed  numbers  to  equivalent  improper 
fractions  : 

1.  13}.    S.  1040^.     6.  17|.     7.  14$.  9.  121f        11.  43f 

2.  27J.    4.  186^.       &  78f-    8-  26A-  10-  17°A-      12-  425A- 

226.  The  value  of  an  improper  fraction  represents  a  quantity 
which  may  be  expressed  by  a  whole  or  a  mixed  number. 

227.  To  reduce  an  improper  fraction  to  a  whole  or  a  mixed  number. 

228.  Example.     Eeduce  if^  to  a  mixed  number. 

o«4  SOLUTION.     Since  5  fifths  equals  1,  134  fifths  must  equal  as 

K         ?       many  times  1  as  there  are  times  6  in  134,  which  are  26|  times. 
Hence,  if*  equals  26|. 

ORAL  EXERCISE 

Convert  into  whole  or  mixed  numbers  : 
1.   -2.      3.   *.      5.       .      7.       -      £       •      **'       -      *&       • 


WRITTEN  EXERCISE 
Convert  into  whole  or  mixed  numbers  : 

J.    ^ft.  7. 

6.   j  8.   ii.         10. 


74  FRACTIONS  [§§  229-234 

229.  To  reduce  a  fraction  to  its  lowest  terms. 

230.  Examples.    1.   Reduce         to  its  lowest  terms. 


J£  as  J.  SOLUTION.    Observing  that  the  terms  of  the  fraction 

T$&  are  divisible  by  3,  first  divide  by  3  and  obtain  as 
a  result  f$.  Observing  that  the  terms  of  the  fraction  f£  are  divisible  by  7, 
divide  by  7  and  obtain  as  a  result  §.  Since  3  and  5,  the  terms  of  the  last  frac- 
tion, are  relatively  prime,  the  reduction  cannot  be  carried  further.  Hence,  ffo 
reduced  to  its  lowest  terms  equals  f  . 

Since  both  numerator  and  denominator  have  been  divided  by  the  same 
numbers,  the  value  of  the  fraction  remains  unchanged  (219). 

8.   Reduce  fJ-J  to  its  lowest  terms. 

SOLUTION.    Being  unable  to  determine  a  common 

Or.  0.  .LI.  =  41.     divisor  of  697  and  779  by  inspection,  find  their  greatest 

697  -f-  41  =  17     common  divisor  (144)  and  obtain  41.    Dividing  the  terms 

779  H-  41  =  19     of  the  fraction  by  41  the  result  is  }$.    17  and  19  being 

relatively  prime,  the  reduction  cannot  be  carried  further, 

WRITTEN  EXERCISE 

Reduce  to  their  lowest  terms  : 

1.  «.         3.  ft.         6.  ^          7.  {if*  9. 

s.  ft        4-  A*       A  HW-       A  «Hf       10. 

231.  In  obtaining  final  results,  aU  proper  fractions  should  be 
reduced  to  their  lowest  terras  and  att  improper  fractions  to  whole 
or  mixed  numbers. 

232.  To  reduce  a  fraction  to  higher  terms. 

233.  Example.     Reduce  f  to  a  fraction  whose  denominator  is  63. 

SOLUTION.    Since  9  is  contained  7  times  in  63,  the  given 

63  -4-  9  =  7.     fraction  may  be  reduced  to  a  fraction  whose  denominator  is 

«  _  05  63,  by  multiplying  both  of  its  terms  by  7.      Multiplying 

both  terms  of  |  by  7,  the  result  is  ||.     ||  has  the  same 

value  as  |  (219). 

In  practice  think  only  of  results.    Thus,  for  the  above  work:  03  -t-  9  =  7. 
6x7  =  35.    Jf. 

234.  Hence  the  following  rule: 

Divide  the  required  denominator  by  the  denominator  of 
the  given  fraction. 


§§234-239]  COMMON  FRACTIONS  75 

Multiply  the  numerator  of  the  given  fraction  by  the  quo- 
tient thus  obtained  and  write  the  product  over  the  required 
denominator. 

The  result  is  tJie  fraction  in  higher  terms. 

ORAL  EXERCISE 

1.  How  many  ninths  in  f  ?  fifteenths  ?  thirtieths  ?  sixtieths  ? 

2.  Change  -|  to  an  equivalent  fraction  having  45  for  its  denomi- 
nator. 

8.   Eeduce  J  to  a  fraction  whose  denominator  is  32  ;  64  ;  128. 

4>  Eeduce  -J  to  sixty-thirds;  to  one-hundred-eighths;  to  ninety- 
ninths. 

5.  If  ^  of  a  number  is  9,  what  is  |  of  the  same  number  ? 

6.  If  ^  of  a  number  is  3,  what  is  ^  of  the  same  number  ? 

7.  How  many  thirty-seconds  in  1  ?  in  \  ?  in  1  ?  in  2£  ?  in  4J  ? 


235.  A  common  denominator  of  two  or  more  fractions  is  any  num- 
ber  which   will   contain  each  of  the  given  denominators  an  exact 
number  of  times. 

236.  The  least  common  denominator  of  two  or  more  fractions  is  the 
least  number  that  will  contain  each  of  the  given  denominators  an 
exact  number  of  times. 

237.  The  least  common  multiple  (134)  of  all  the  denominators 
of  the  given  fractions  is  the  least  common  denominator. 

238.  To  replace  two  or  more  fractions  by  two  or  more  equivalent 
fractions  having  the  least  common  denominator. 

239.  Example.    Eeduce  f  ,  -J-,  and  ^  to  equivalent  fractions  having 
a  common  denominator. 

~  T5"*  SOLUTION.     The  least  common  multiple  of  the 


-R-  —  ITS--     given  denominators  is  found  to  be  36.     Using  36 


2-1-3     -jig-  =  -£2g-.     as  the  least  common  denominator  reduce  each  of 

vx  Q  ,x  o  v,  Q       Q£       the  given  fractions  to  thirty-sixths,  as  explained 
XoX^Xo  =  oo.  .      .»  .       „ 


L.C.M.  =  36.          m233' 


76  FRACTIONS  [§§  240-244 

240.    Hence  the  following  rule  : 

Find  the  least  common  multiple  of  each  of  the  given  de- 
nominators for  the  least  common  denominator,  and  proceed 
as  in  234. 

ORAL  EXERCISE 

By  inspection,  reduce  to  equivalent  fractions  having  the  least 
common  denominator  : 

1-   f>  f  *•    i    f.  5.   i    i    1,  i  7.   i    i,  i    TV 

2.  f  |.          4-  I,  f          e.  i,  i,  ^  A-        *  i  i  T^  i- 

WRITTEN  EXERCISE 

Reduce  to  equivalent  fractions  having  the  least  common  denomi- 
nator : 


ADDITION  OF  FRACTIONS 

241.  Similar  fractions  are  fractions  having  the  same  denominator 
or  unit  value. 

242.  In  order  that  fractions  may  be  added,  they  must  be  similar 
and  parts  of  like  units. 

243.  The  denominator  is  the  namer  of  the  fraction  ;  hence,  simi- 
lar fractions  are  analogous  to  like  numbers. 

244.  To  add  similar  fractions, 

Add  tfo  numerators. 

Place  the  sum  over  the  common  denominator,  and  reduce 
the  result  to  its  simplest  form. 

ORAL  EXERCISE 
By  inspection,  find  the  sums  of: 

i-  i  +  i  +  f  +  |  +  i  +  i  +  i         4.  tV  +  A  +  A  +  TV 

«.  *  +  f  +  5  +  4  +  *  +  i}  +  f      5.  A  +  A  +  A+A. 

•      *       +     +     +     +  4    +  6.       .  +    ,  +       +      . 


§§244-247]  COMMON   FRACTIONS  77 

By  horizontal  addition,  find  the  sums  of: 

7.  3  pieces  of  print  containing  412,  241,  403  yards,  respectively. 

NOTE,  lii  the  dry  goods  business  fourths  (quarters)  are  very  common  frac- 
tions, and  they  are  usually  written  without  denominators  by  placing  the  numera- 
tors slightly  above  the  integers.  Thus,  411  =  41£,  412  =  41|  (41  fc),  and  413  =  41|. 

8.  3  pieces  wash  silk  containing  421,  451, 431  yards,  respectively. 

9.  3  pieces  duck  containing  372,  412,  45  yards,  respectively. 

10.  4  pieces  cashmere  containing  421,  451,  461,  421  yards,  respec- 
tively. 

245.  To  add  fractions  not  having  a  common  denominator. 

246.  Examples.     1.   Find  the  sum  of  f  and  f . 

5      3      40-4-21  SOLUTION.      Since   the    given    fractions  are  not 

«  ~f~  Q  =       f-jT* similar,  reduce  them  to  equivalent  fractions  having 

the  least  common   denominator  as  in  240.      Then 

=5  —  =  l-g5_,     add  the  numerators  and  place  the  sum,  61,  over  the 

56  least  common  denominator,  66.     $  =  *&• 

8.  Find  the  sum  of  47J,  16|,  and  17|. 

Eighths 

47  J  4                       SOLUTION.    By  inspection,  the  common  denominator 

16}  6  of  the  fractions  is  found  to  be  8.    The  sum  of  the  frac- 

17J-  5  tions  is  then  1$,  which,  added  to  the  sum  of  the  whole 

81 J  15  numbers,  gives  81 J,  the  required  result 

T        * 

247.  Hence  the  following  rule : 

Reduce  the  given  fractions  to  equivalent  fractions  having 
the  least  common  denominator t  and  add  as  in  244. 


WRITTEN  EXERCISE 

Find  the  sums  of  the  following  fractions : 

i.  f ,  4,  f  •       4-  f,  A,  *        7.  A,  j,  A.        10.  |,  4,  |. 
*•  f .  t»  A-      s.  &,  I,  A-       *  »»  f  f  «•  A' t  A 

&  f»  A,  A-     «•  A.  1. 1-         »•  *»  A>  A-        ^-  I.  T,  1- 


78  FRACTIONS  [§§  248-250 

SHORT  METHODS 

248.  When  the  numerators  of  any  two  fractions  are  alike,  the 
work  of  addition  may  be  performed  as  shown  in  the  following 
examples. 

249.  Examples.     1.   Find  the  sum  of  -J-  and  -fa. 

SOLUTION.    The  common  denominator  is  2  x  17,  or  34. 

£  -|-  -J^  =  ^J.    Since  the  numerator  of  each  of  the  fractions  is  1,  the 

numerator  of  the  first  equivalent  fraction  having  a  common 

denominator  34  is  equal  to  the  denominator  of  the  second  fraction ;   and  the 

numerator  of  the  second  equivalent  fraction  having  a  common  denominator  34  is 

equal  to  the  denominator  of  the  first  fraction.    2  +  17  =  19;  hence  the  sum  of  the 

given  fractions  is  found  to  be  £f . 

8.  Find  the  sum  of  -f-  and  f . 

SOLUTION.     The  common  denominator  is  7  x  9,  or  63. 
•2-  -|-  -3-  =  |-£.      If  the  numerator  of  each  of  the  given  fractions  were  1,  the 
sum  of  the  numerators  in  the  equivalent  fractions  having  a 
common  denominator  63  would  be  16  (7  +  9).     But  the  numerator  of  the  given 
fractions  is  2  ;  hence  the  sum  of  the  numerators  of  the  equivalent  fractions  having 
a  common  denominator  63,  is  32  (2  x  16).     Therefore,  $  +  f  =  if. 
In  practice  think  only  of  results.    Thus,  7  +  9  x  2  =  32  •  ||. 

ORAL  EXERCISE 
By  inspection,  find  the  sum  of : 

1.  1  +  1.       8.  A  +  f      16.     f +  f     M.     |-f  f       «9. 

2.  *  +  £.        9.     i-f-J.      18.     f  +  f.      23.     f  +  A      SO. 

3.  i  +  i.      10.     i  +  i-     17.  A  +  f     **     *  +  *     91.     f +  A- 

4.  A  +  *«      11-     i  +  i-      18.  A  +  f      **•     1  +  1-        «*•     t  +  f 

5.  A  +  t     ^     i  +  J*     ^     *  +  i     ^     f  +  f- 

^.  A+^    ^    *  +  £    ^  A+f    ^    l  +  f     ^    f 
r.  A  +  £    ^    *  +  f    ^-  A  +  f    ^  A-  +  *- 

In  the  following  problems  add  the  first  two  fractions,  and  to  the 
sum  add  the  other  fraction. 

ss.  i  +  t  +  f  as.  t  +  t+TV  40-  l  +  ?  +  rt- 

«••  i+i-t-A-         ».  i+i+A>  •#•  ?+f+tt- 

250.  The  business  man's  fractions  are  usually  of  the  simplest 
sort,  and  ability  to  add  them  rapidly  is  of  the  utmost  importance.    In 
a  great  many  cases  the  least  common  denominator  can  be  determined 
by  inspection  and  the  fractions  added  as  rapidly  as  whole  numbers. 


§251]  COMMON   FRACTIONS  79 

251.   Examples.     1.   Find  the  sum  of  -J-,  J,  £,  and  ^ 

SOLUTION.     By  inspection,  find  the  least  com- 

1,1,1,  _L__  1 5      mon  denominator  to  be  16.    Reducing  each  fraction 
r  1 6      T5*     to  sixteenths  at  sight,  and  adding,  say  or  think,  1, 
3,  7,  15,  ft. 

2.  Find  the  sum  of  f ,  T^,  £,  and  J. 

SOLUTION.     By  inspection,  find  the  least 
yf =1^-.     common  denominator  to  be  16.     4,  6,  7,  19, 
tt.  or  IrV 


ORAL  EXERCISE 

By  inspection,  find  the  sums  of  the  following  problems  : 

1.       2.       8.     4-       5.        6.       7.      8.       9.      10.     11.     12. 

iiifiViiilif* 
A  t  i  i  A  i  f  A  *  i  f  * 
AAAAI  I  i  t  A  V  A  f 

_lJLil_L±AlJiJLA±    I 

The  above  may  be  used  as  an  "open-book"  exercise,  different  students 
being  required  to  announce  at  sight  the  least  common  denominator,  and  then 
the  successive  steps  necessary  to  arrive  at  a  total.  Students  should  be  drilled  on 
exercises  similar  to  the  above  until  they  can  add  the  fractious  given  as  rapidly  as 
they  can  whole  numbers. 

WRITTEN  EXERCISE 

Copy  or  write  from  dictation  and  add  the  following  problems. 
Add  the  fractions  as  explained  in  251. 


1040J 
1620J 
1342| 
1647| 
1842^ 
16211- 
1831J 


80  FRACTIONS  [§§  252-250 

SUBTRACTION  OF  FRACTIONS 

252.   In  order  that  fractions  may  be  subtracted  they  must  be 
similar  and  parts  of  like  units. 

DRILL  EXERCISE 

0.  478  yards -28  yards  =?    5.     f-f    =?    8.     fi  -  if   =? 
S.  195  acres  -  88  acres  =  ?    6.     J-f=?^ttJ- 


253.  To  subtract  similar  fractions, 

Find  the  difference  between  the  numerators  of  the  given  fractions 
write  the  result  over  the  denominator. 

254.  To  subtract  fractions  not  having  a  common  denominator. 

255.  Examples.     1.   Subtract  £  from  £. 

7    _  _   o-i  SOLUTION.     Since  only  similar  fractions  and  parts  of  like? 

"  units  may  be  subtracted,  reduce  the  given  fractions  to  equiva- 

§  **  _      lent  fractions  having  a  common  denominator.    |  =  f £ ;  |  =  if. 

A     tt-H  =  A- 

Hence,  J-|  =  A- 

0.  From  16J  take  12|. 

SOLUTION.     Reduce  the  fractions  to  a  common  denomi- 
•"i  "  l^rf      nator.     Since  we  cannot  take  ^  from  ^,  we  take  1  from  16, 
12|  =  12^      change  it  to  ^f,  and  add  it  to  ^,  making  ^f .    Then  ^|  -  ^ 
=  ^  ;  and  15  -  12  =  3. 
Hence,  16|  -  12|  = 


256.   Therefore  the  following  rule : 

Reduce  the  given  fractions  to  equivalent  fractions  having 
the  least  common  denominator  and  subtract  as  in  253. 

ORAL  EXERCISE 

Find  the  values  of  the  following : 

1.  lf-f        S.  7f-lf          5.  12f-3f.        7.  ll^-Sf 

&  3J-f        4.  25J-14f      6.   17|J-f        8.   120  -  56f£. 


256-258]  COMMON   TRACTIONS  81 

WRITTEN  EXERCISE 
Find  the  difference  between : 

1.  240f  and  89£.        4.   ll^J  and  21f        7.   1050f£  and  2020| 

2.  |  J  and  |f.  5.   104^  and  84f        8.   79f  and  49J. 
5.   14|and21f          6.   9Jf  and  21.  9.   541 1  and  29 J. 

10.   From  216J  acres  of  land,  lots  of  21  acres,  16-f  acres,  26||  acres, 
acres,  and  63 J  acres  were  sold.      How  many  acres  remained 
unsold  ? 

SHORT  METHODS 

257.  When  the  numerators  of  any  two  fractions  are  alike,  the 
work  of  subtraction  may  be  shortened,  as  shown  in  the  following 
examples. 

258.  Examples.    1.  From  |  take  -3^. 

SOLUTION.     The  common  denominator  is  17  x  9,  or 

•J  —  TT  =  ITS*      153.    The  numerator  of  the  first  equivalent  fraction  hav- 
ing a  common  denominator  153  is  1  x  17,  or  17.    The 

numerator  of  the  second  equivalent  fraction  having  a  common  denominator  153 
is  1x9,  or  9.  Hence  17,  the  denominator  of  the  subtrahend,  minus  9,  the 
denominator  of  the  minuend,  written  over  the  common  denominator,  153,  equals 
the  required  result,  or  jl^. 

2.   From  |  take  T\. 

SOLUTION.    The  common  denominator  is  91.    13  —  7 
fy  —  -fa  =  -J-J.       x  2  =  12,  which  write  over  the  common  denominator  91. 

ORAL  EXERCISE 

By  inspection,  find  the  value  of: 

IS.     2__2T.      19f  !_|.      $5. 

14.    l-£  20.  f-^.  26. 

5.  £_i,        9,  -2—^-.      15.  A— A-  #•*•  f  — fr  #7-  124|— 13f 

£  |-^.     10.  |-f        16.    f-tV  22.  f-f.  28.  64|-52f 

5.  |-f       11.  f— &.      17.    f-f.  23.  f-J.  29.  83i-72£. 

0.  £-.£.       j&  f—ft.      7<5.     |-f  ^.  f-f.  m  89|-72f. 


82  FRACTIONS  [§§  258-260 

MULTIPLICATION  OF  FRACTIONS 
DRILL  EXERCISE 

1.  If  1  pound  of  sugar  is  worth  5^,  how  much  are  5  pounds 
worth  ? 

2.  If  1  pound  of  tea  is  worth  $  ^,  how  much  are  3  pounds  worth  ? 
8.   If  12  men  earn  $48,  how  many  dollars  does  one  man  earn? 

4.  What  is  |  of  48  ?  6.   What  is  f  of  15  ? 

5.  48x^  =  ?  7.   15  xf  =? 

259.  To  multiply  a  fraction  by  a  whole  number,  a  whole  number  by 
a  fraction,  or  a  fraction  by  a  fraction. 

260.  Examples.     1.  f  x  4  =  ? 

(a)  SOLUTIONS.     («)  Multiplying  the  numerator  of 

3  3x43  a  fraction  multiplies  that  fraction  (219).     Hence, 

•;;  X  4  =         P  =  —  =  1-J-.      to  multiply  f  by  4,  multiply  the  numerator  3  by  4 

and  divide  the  result  by  the  denominator  8  as  shown 

in  (a).    Or, 

(P)  (6)  Dividing  the  denominator  of  any  fraction 

3      ^  __     3      __  3  __  ^  i       multiplies  that  fraction  (219).    Hence,  to  multiply 

8  8-5-42  f  by  4  divide  the  denominator  8  by  4  and  write  the 

numerator  over  the  quotient  as  shown  in  (6).    Or, 

/c\  (c)  Arranging  (a)  in  another  convenient  form 

3  3x43  *or  canceuation  we  iiave  the  Process  (c). 

—  X  4  =  - — 2  =  -  =  1-J-.  Multiplying   the  numerator  and   leaving  the 

P  denominator  unchanged  as  in  (a),  the  number  of 

parts  taken  has  been  multiplied,  and  the  value  of 
each  part  left  the  same.    Hence,  the  whole  fraction  has  been  multiplied. 

Dividing  the  denominator  and  leaving  the  numerator  unchanged  as  in  (6), 
the  size  of  each  part  has  been  multiplied  and  the  number  of  parts  left  the  same. 
Hence,  the  whole  fraction  has  been  multiplied. 

A  whole  number  may  be  expressed  in  fractional  form  by  writing  1  for  its 

denominator.     Hence,  -  x  4  =          ,  and  the  process  is  indicated  in  convenient 
form  for  cancellation. 

2.  What  will  5  dozen  oranges  cost  at  9  f  per  dozen  ? 

3      «  SOLUTION.    Since  1  dozen  oranges  cost  $f,  6  dozen  will  cost 

«  x  1=         5  times  $|,  which,  by  cancellation,  as  shown  in  the  margin,  is 
equal  to  $3. 


§§260-261]  COMMON  FRACTIONS  83 

8.  If  1  mat  of  coffee  cost  $24,  what  will  f  of  a  mat  cost  ? 

(a)  SOLUTIONS,    (a)  f  of  $24  =  }  of  3  x  $24.    3  x  $24  =  $72. 

24  x  £  =  10 2.     $  of  $72  ($72  -:-  7)  =  $  lOf.    Hence,  f  of  a  mat  of  coffee  at 

$24  a  mat  will  cost  $  10  £.    Or, 

(&)  (6)  Since  1  mat  of  coffee  cost  $  24,  f  of  a  mat  will  cost 

24  -J-  7  =  3f        f  of  $24.    f  of  $24  is  equal  to  3  times  }  of  $24.    }  of  $24 
38X3  =  102.      ($24-*-7)  =  $3f    3  times  $3?  =  10?. 

Solution  (6)  is  shorter  than  solution  (a)  when  the  denomi- 
nator of  the  divisor  is  a  factor  of  the  whole  number. 

4.  If  1  pound  of  tea  cost  $|,  what  will  |  of  a  pound  cost  ? 

9  :.  ^  K  SOLUTIONS,  (a)  If  1  pound  of  tea  cost  $|,  $  of  a  pound 

*  x  -  =  —  =  —  •  ^H  cost  §  of  $f-  §  of  $|  is  equal  to  2  times  $  of  $f. 

8  3  24  12  |  of  $  f  equals  $  £.  2  x  $  &  =  $  J$  =  $  T\.  Hence,  if 

/JA  1  pound  of  tea  cost  $|,  |  of  a  pound  will  cost  $-^j.  Or, 

M      n       K  (^)  By  cancellation  the  operation  is  a  little  shorter 

—  X  J  =  —  •        and  the  result  appears  in  its  lowest  form, 
p     o     U 

^.  Find  the  product  of  7J  X  3j. 

j~      ^      ww  SOLUTION.    Reduce  the  mixed  numbers  to  impropei 

4f  X  —  =  —  =  27 J.    fractions  and  proceed  as  explained  in  Example  4. 

2       p       A 

261.  From  the  foregoing  solutions  the  following  general  rule 
may  be  derived: 

Express  tlw  whole  or  mixed,  numbers  as  improper  frac- 
tions. Cancel  all  equivalent  factors  from  the  numerators 
and  denominators. 

Find  the  product  of  the  remaining  numerators  for  the 
numerator  of  the  resulting  fraction,  and  the  product  of 
the  remaining  denominators  for  the  denominator  of  the 
resulting  fraction. 

NOTE.    The  same  rule  holds  good  for  finding  the  product  of  more  than  two 

fractions. 

ORAL  EXERCISE 
Find  the  value  of : 

1.  JoflS.  8.  fxa  5.  }of27.  7.  H°f45. 

2.  18x|.  4-  |x4.  6.  |of6.  8. 


84  FRACTIONS  [§  261 

P.  Find  the  cost  of  20  yards  of  cashmere  at  $f  a  yard. 

10.  If  1  yard  of  silk  is  worth  $T7¥,  what  are  8  yards  worth  ? 

11.  Required,  the  cost  of  150  yards  of  muslin  at  $£  a  yard. 

12.  What  will  2i  pounds  of  sugar  cost  at  3^  per  pound  ? 
18.  What  will  2±  pounds  of  beef  cost  at  6|^  per  pound  ? 
14.  What  will  |  of  a  pound  of  tea  cost  if  1  pound  cost  $  -f  ? 

#?.  John  was  given  -J-  of  a  farm  and  James  f  as  much.    What 
part  had  James  ? 

16.  If  \  of  a  stock  were  lost  by  fire  and  the  remainder  sold  at  f 
of  its  cost,  what  part  of  the  first  cost  was  received  ? 

17.  Divide  21  into  2  parts,  one  of  which  shall  be  f  of  the  other. 

18.  Divide  60  into  2  parts,  one  of  which  shall  be  {•  of  the  other. 

19.  So  divide  $  150  between  A  and  B  that  A's  part  may  be  J  of  B's. 

20.  Tea  costing  $  f  per  pound  is  sold  for  f  of  its  cost.    For  what 
price  per  pound  is  the  tea  sold  ?    What  is  the  loss  per  pound  ? 

WRITTEN  EXERCISE 

Fmd  the  product  of : 

1.  ^x85.         &  ^  x!2.          6.  A^xia          7.  if*x40. 
*.  |Jx8.  ^.   J^x9.  6.  VxlL          &  ^f  x28. 

P.  £xf  xf  xfxf  xf  xfxlOa? 
m  »x«x5xAx»xi-f 

If.  What  will  be  the  cost  of  7£  tons  of  hay  at  f  of  $15  per  ton  ? 
12.  Find  the  cost  of  7J  pounds  of  beef  at  8^  per  pound. 
18.  At  $2£  per  day,  how  much  will  a  man  earn  in  17£  days  ? 

14.  Paid  $^  for  some  stationery  and  |  as  much  for  some  pens 
and  ink.     How  much  did  I  pay  for  both  ? 

15.  A  having  $750  invested  -J-  of  it  in  insurance  and  paid  }  of 
the  remainder  for  a  horse.     How  much  did  he  have  left  ? 

16.  A  owning  f  of  a  mill  sold  f  of  his  share  to  B.    What  part  of 
the  whole  mill  did  he  still  own  ? 


§§262-265]  COMMON   FRACTIONS  85 

SHORT  METHODS 

262.  To  multiply  together  two  mixed  numbers  when  the  integers  are 
alike  and  the  fractions  are  |- 

263.  Example.    What  will  8£  yards  of  lining  cost  at  8^  per 
yard? 

8^  SOLUTION.     The  sum  of  J  of  each  of  two  like  numbers  is  equal  to 

gl          either  of  the  numbers.    Since  8  multiplied  by  £  added  to  £  of  8  is 
797^       equal  to  8,  in  finding  the  product  of  8£  by  8|,  we  may  say,  9x8 
•f  $  of  £  =  72£,  the  required  result. 

ORAL  EXERCISE 
By  inspection,  find  the  value  of  : 

1.  2%  yards  at  $  2|.     4.  3£  yards  at  $  3  J.  7.   6£  pounds  at  6J  £ 

2.  7-|  barrels  at  $7f   5.   8J  pounds  at  8^.  8.  11|  acres  at  $  11  J. 
5.   5|-  yards  at  $  5J.      5.   9J  pounds  at  9j£  P.   12£  dozen  at  $  12  J. 


264.   To  multiply  together  any  two  mixed  numbers  when  the  frac- 


tions are  -. 


265.   Examples.    1.  Find  the  cost  of  120J  yards  of  lining  at  6%tf 
per  yard. 


SOLUTION,    j  of  120  -1-  6  times  \  is  equal  to  \  of  120  +  6. 
\  of  126  =  63,  which  write  as  shown  in  the  margin.     \  of  \  -  £, 
63J  which  write  as  shown  in  the  margin.    6  times  120  =  720.    Adding 

720  ti*6  Partial  products,  the  result  is  7.83£.     The  required  result  is, 

/-r  00^  therefore,  $7.83. 

2.  Find  the  cost  of  87  J  pounds  of  crackers  at  S^  per  pound. 


_ 

SOLUTION.     £  of  87  +  8  =  47£,  which,  added  to  \  of  \  =  47f  . 
87  x  8  =  696.    Add,  and  the  result  is  7.43|,  or  $  7.44. 

7.43|' 

Observe  that  In  dividing  numbers  by  2  there  can  never  be  a  remainder 
of  more  than  1. 

Also  that  in  multiplying  together  any  two  numbers  ending  in  £,  the  fraction 
in  the  resulting  product  must  be  either  \  or  J. 


86  FRACTIONS  [§§  266-267 

266.    Therefore  the  following  rule  : 

If  the  sum  of  the  integers  is  even,  write  f  in  the  resulting 
product.  If  not,  write  \  in  resulting  product. 

Find  one  half  of  the  sum  of  the  integers  and  to  the  result 
add  the  product  of  the  integers. 

ORAL  EXERCISE 

By  inspection,  find  the  value  of  : 

1.  9£  yards  at  6j£         4.  161  yards  at  10i£    7.  351  yards  at 

2.  4|-  pounds  at  9i£      5.  20£  yards  at  8j£      8.  121  yards  at 

S.  12J-  pounds  at  7^.    6.  321  yards  at  2  J/      P.  211  pounds  at  4j£ 

WRITTEN  EXERCISE 

Find  the  value  of  : 

1.  162$.  x  16J.        £  123J-  X  51.  7.  120J  x  4£.  m  204|  x  7|. 

&  144J  x  3J,          5.  60|  x  121  8.  H5J  X  51  ^.  2151  x  8i 

x  4f  (5.  90J  x  3|.  P.  16|  x  5J.  12.  1101  x  8J. 


Mixed  numbers  ending  in  £,  |,  etc.,  may  be  multiplied  together  in  prac- 
tically the  same  manner  as  shown  in  the  foregoing  explanations. 

DIVISION  OP  FRACTIONS 
DRILL  EXERCISE 

1.  If  8  hats  cost  $24,  what  will  1  hat  cost? 

2.  If  |  of  an  article  cost  $  24,  what  will  \  of  an  article  cost  ? 
8.   If  5  barrels  of  apples  cost  $  15,  what  will  7  barrels  cost  ? 

4.  If  f  of  an  acre  of  land  cost  $  15,  what  will  £  acre  cost  ? 

5.  If  3  pounds  of  tea  cost  $  1.05,  what  will  5  pounds  cost  ? 

6.  If  f  of  a  pound  of  tea  cost  21  £  what  will  f  of  a  pound  cost  ? 

7.  If  5  dimes  will  purchase  1  pound  of  tea,  how  many  pounds 
will  $  4  purchase  ? 

8.  If  i^  of  a  pound  of  tea  cost  25^,  what  will  4  pounds  cost? 

9.  If  4  pounds  of  sugar  cost  $  \,  what  will  1  pound  cost  ? 
10.   If  |  of  a  yard  of  cloth  cost  $  |,  what  will  1  yard  cost? 

267.   The  reciprocal  of  a  whole  number  is  1  divided  by  that  number. 
the  reciprocal  of  3  is  J. 


§§268-272]  COMMON  FRACTIONS  87 

268.  The  reciprocal  of  a  fraction  is  1  divided  by  that  fraction. 

269.  Example.     Find  the  reciprocal  of  £. 

SOLUTION.     In  1  there  are  6  fifths.     \  is  contained  In  6  fifths  6  times,     f  Is 

4  times  £.     Therefore,  |  is  contained  in  f  \  as  many  times  as  $.     £  of  6  is  J ; 
the  reciprocal  of  f  equals  f.     Hence, 

I  divided  by  any  fraction  is  the  fraction  inverted. 

270.  \  is  the  reciprocal  of  4,  and  4  is  the  reciprocal  of  £.    Hence, 

27&#  dividend,  multiplied  by  the  reciprocal  of  the  divisor, 
is  equal  to  the  quotient. 

271.  To  divide  a  fraction  by  a  whole  number,  a  whole  number  by  a 
fraction,  or  a  fraction  by  a  fraction. 

272.  Examples.     1.   Divide  f  by  2. 

(a)  SOLUTIONS,     (a)   Dividing  the  numerator  (divi 

4.  4  _._  2      2  dend)  divides  the  fraction  (219).     Hence,  divide  the 

K~*~    = -     =-•  numerator  of  the  fraction  f  by  2  and  the  quotient 

is§.    Or, 

(6)  Multiplying  the  denominator  (divisor)  divides 
^  .  o_     ^     _  4  _2     the  fraction  (219).     Hence,  multiply  the  denomi- 

5  5x2      10      5*    nator  of  the  fraction  f  and  the  quotient  is  §.     Or, 

(c)  (c)  The  dividend  multiplied  by  the  reciprocal 

2  of  the  divisor  is  equal  to  the  quotient  (270).    The 

4_jL_2==|xl==?-  reciprocal  of  2  is  £.    Therefore,  £  divided  by  2  is 

5  '          5      %     5  equal  to  |  x  }.    |  x  *  =  f 

#.   If  2  pounds  of  coffee  cost  $  f ,  what  will  1  pound  cost  ? 

o 

~|      .,      o  SOLUTION.     If  2  pounds  of  coffee  cost  •$£,   1 

2  X  -  =  -.  pound  wijl  cost  i  of  f ,  or  $f. 

5     jfi     o 

5.   At  $|-  per  yard,  how  many  yards  of  cloth  may  be  purchased 

forfU? 

SOLUTIONS,     (a)   Since  the  denominator  names 
or  tells  the  size  or  value  of  the  parts  taken,  similar 

-.  .  _._  7  _  112  _._  7  fractions  may  be  divided  as  concrete  whole  numbers. 

8  ~~    8        8  In  concrete  whole  numbers  the  operation  of  division 

_  H2  -s-  7  =  16  is  performed  without  regard  to  the  unit  named,  so 

in  dividing  similar  fractions  the  denominator  may 

be  ignored.     Reducing  14  to  the  same  value  as  |,  the  result  is  ^p.     H2  + 1  bein& 
analogous  to  -£—  -*.  — £— ,  divide  112  by  7  and  obtain  16. 

dollar^       dollars' 


88  FRACTIONS  [§§  272-273 

In  a  critical  analysis  of  the  above  problem  say  in  general :  Since  1  yard  cost 
$£,  as  many  yards  may  be  purchased  for  $14  as  $  is  contained  times  in  14. 
14  =  i|2,  and  £  is  contained  in  ^  16  times.  Hence,  16  x  1  yard,  or  16  yards, 
may  be  purchased  for  $  14.  Or, 

x,v  (6)  Since  1  yard  cost  $f,  as  many  times  1  yard 

may  be  purchased  for  $  1  as  $  $  is  contained  times  in 

~.      ~  $1,  or  f  times  (the  reciprocal  of  £).     f  x  1  yard  =  f 

14  -5-  E£  X  -  =  16.        yards.     Since  for  $  1  f  yards  may  be  purchased,  for 

1       /•  $14  14  times  f  yards,  or  16  yards,  may  be  purchased. 

4.  If  1  yard  of  cloth  cost  $  ^,  how  many  yards  may  be  purchased 

forff? 

SOLUTIONS,     (a)  Since  1  yard  cost  $£,  as  many 

(a)  yards  may  be  bought  for  $f  as  ^  is  contained  times 

}  -«-  -£  =  -j^j-  -J-  -^j-  =       in  f .     |  =  T9s  ;  £  =  ^j.    4  is  contained  in  9  2£  times. 
9  _._  4  =  21  Hence,  2£  x  1  yard,  or  2£  yards,  may  be  purchased. 

Or, 

(6)  Since  $£  is  contained  in  $1  f  times  (the 

.,,.  reciprocal  of  |),  3  yards  of  cloth  may  be  purchased 

i      8  Vs      9      91        for  $1.     If  3  yards  may  be  purchased  for  $1,  f  of  3 

f  ~*~7=¥x  T=f = ^¥*      yards  may  be  purchased  for  $|.    Hence  the  simple 


273.   Therefore  the  following  rule  : 

Multiply  the  dividend  ~by  the  reciprocal  of  the  divisor. 

NOTE.     Reduce  mixed  numbers  to  improper  fractions  before  applying  the 
rule. 

ORAL  EXERCISE 
Find  the  value  of  : 


1.     6-*-£  &  24-*-}.  6.  10  +f  7.  }-*-6. 

£  12-*-}.  4.  15-*-£.  ft   25  +  !.  *  t^3- 

P.  If  4  pounds  of  coffee  cost  $  f  ,  what  will  1  pound  cost  ? 

70.  If  |_i.  Of  a  farm  be  grain  land  and  evenly  divided  into  3 
6  elds,  what  part  of  the  farm  will  each  field  contain? 

11  Divide  9  by  |;  12  by  1J;  \  by  f  ;  2|  by  ^. 

J&  What  part  of  1  is  |?  of  2  ?  of  3?  of  4?  of  5?  of  10? 

18.  What  part  of  9  is  4?  of5?  of  6  ?  of  7?  of  11  ?  of  17? 


§273]  COMMON  FRACTIONS  89 

WRITTEN  EXERCISE 

L  If  |  of  an  acre  of  land  sells  for  $45,  what  will  1  acre  sell  for 
at  the  same  rate  ? 

2.  A  farm  of  471  acres  is  divided  into  shares  of  94J  acres  each. 
How  many  shares  are  there  ? 

S.  A  church  collection  of  $  232  was  divided  among  poor  families, 
to  each  of  which  was  given  $5J.  How  many  families  shared  the 
bounty  ? 

4.  When  potatoes  are  worth  $  f  per  bushel  and  apples  $  f  per 
bushel,  how  many  bushels  of  potatoes  would  pay  for  a  load  of  apples 
measuring  30  bushels  ? 

5.  A  woman  buys  f  of  a  cord  of  wood  worth  $  6|  per  cord  and 
pays  for  it  in  work  at  $  £  per  day.     How  many  days  must  she  work 
to  make  full  payment  ? 

6.  A  dealer  paid  f  of  $78f  for  f  of  25  cords  of  wood.     What 
was  the  cost  per  cord  ? 

7.  if  _^.  of  a  farm  of  67£  acres  be  divided  into  63  village  lots, 
what  part  of  an  acre  will  each  lot  contain  ? 

8.  1760  bushels  of  wheat  were  put  into  sacks  containing  2J 
bushels.     How  many  sacks  were  there  ? 

9.  At  $  -J  per  day,  how  long  would  it  take  to  earn  $  15  f  ? 

10.  How  many  fields  of  9|  acres  can  be  made  from  a  farm  con- 
taining 125  J-  acres  ? 

WRITTEN  REVIEW 


1.  From  the  sum  of  %  and  5J  take  the  difference  between 
and  21. 

2,  Divide  into  six  equal  parts  the  product  of  11^  multiplied 


8.  An  estate  is  so  divided  among  A,  B,  and  C  that  A  gets  f, 
B  ^,  and  C  the  remainder,  which  was  $4200.  What  is  the 
amount  of  the  estate  ? 

4.  If  14  bushels  of  apples  can  be  bought  for  $3|,  how  many 
bushels  can  be  bought  for  f  f  ? 

5.  A  woman  having  $  1  gave  f  of  it  for  coffee  at  33^  per  pound. 
How  many  pounds  did  she  buy  ? 

6.  Having  bought  |  of  a  ship,  I  sold  f  of  my  share  for  $12,000. 
What  was  the  value  of  the  ship  at  that  rate  ? 


90  FRACTIONS  [§273 

7.  What  must  be  the  amount  of  an  estate  it,  when  it  is  divided 
into  three  parts,  the  first  part  is  double  the  second,  the  second  double 
the  third,  and  the  difference  between  the  second   and  the   third 
is  $7500? 

8.  Having  paid  $  119  for  a  watch  and  chain,  I  discover  that  the 
cost  of  the  chain  was  only  -^  the  cost  of  the  watch.     What  was  the 
cost  of  the  watch  ? 

9.  An  estate  valued  at  $  120,000  was  so  distributed  that  A  re- 
ceived ^ ;  B  ^  of  the  estate  more  than  A ;  C  as  much  as  A  and  B 
together,  less  $600;  and  2  charities  the  remainder  in  equal  parts. 
How  much  did  each  charity  receive  ? 

10.  A  painter  worked  17^  days,  and  after  expending  ^  of  his 
wages  for  board  had  $  15  left.     How  much  did  he  earn  per  day  ? 

11.  A  mechanic  worked  21f  days,  and  after  paying  his  board 
with  |  of  his  earnings  had  $  66 j-  left.    How  much  did  he  earn  per 
day? 

12.  If  \  of  the  trees  of  an  orchard  are  apple,  J  peach,  J-  pear, 
•^  plum,  and  the  remaining  21  trees  cherry,  how  many  trees  in  all  ? 

IS.  A,  B,  and  C  rented  a  pasture  for  $37.  A  put  in  3  cows  for 
4  months,  B  5  cows  for  6  months,  and  G  8  cows  for  4  months.  How 
much  had  each  to  pay  ? 

14-  A  farmer  sold  two  cows  for  $  75,  receiving  for  one  cow  only 
{  as  much  as  for  the  other.  What  was  the  price  of  each  ? 

15.  After  selling  450  horses  a  dealer  had  -f-  of  his  stock  remain- 
ing.   How  many  had  he  at  first  ? 

16.  If  8  horses  consume  4£  bushels  of  oats  in  3£  days,  how  many 
bushels  will  12  horses  consume  in  the  same  time  ? 

17.  A  and  B  can  do  a  piece  of  work  in  10  days  which  A  alone 
can  do  in  18  days.     In  what  time  can  B  alone  do  the  work  ? 

18.  John  and  Calvin  agree  to  build  a  wall  for  $  86.     If  Calvin 
can  work  only  |-  as  fast  as  John,  how  should  the  money  be  divided  ? 

19.  What  is  the  length  of  a  pole  that  stands  £  in  the  sand,  -J  in 
the  water,  and  25 \  feet  above  the  water  ? 

SO.  A.  colt  and  cow  cost  $124.  If  the  colt  cost  $4  more  than 
It  times  the  cost  of  the  cow,  what  was  the  cost  of  each  ? 


§  273]  COMMON   FRACTIONS  91 

21.  A  tree  84  feet  high  was  so  broken  in  a  storm  that  the  part 
standing  was  f  of  the  length  of  the  part  broken.     How  many  feet 
were  standing  ? 

22.  A  farmer  has  f  of  his  sheep  in  one  pasture,  f  in  another, 
and  the  remainder  of  his  flock,  72  sheep,  in  a  third  pasture.    How 
many  sheep  has  he  ? 

28.  For  a  horse  and  carriage  I  paid  $540.  What  was  the  cost 
of  each,  if  the  cost  of  the  carriage  was  1 J  times  the  cost  of  the  horse  ? 

24'  Peter  can  do  a  piece  of  work  in  12  days,  and  Charles  in  15 
days.  How  many  days  will  it  require  for  its  completion  if  they 
both  join  in  the  work  ? 

25.  A  can  do  a  piece  of  work  in  21  days,  B  in  18  days,  and  C  in 
15  days.    In  how  many  days  can  the  three  working  together  per- 
form the  work  ? 

26.  John  and  his  father  have  joint  work  which  they  can  do, 
working  together,  in  25  days.     If  it  requires  60  days  for  John  work- 
ing alone  to  complete  the  work,  how  many  days  will  it  require  for 
the  father  to  complete  it  ? 

27.  A,  B,  and  C  together  have  $  2520.    C  has  twice  as  much  as 
B,  who  has  J  as  much  as  A.     How  much  has  each  ? 

28.  A  farmer  bought  3  farms  of  240  acres  each  at  $llf  an  acre. 
He  built  three  barns  at  a  cost  of  $1245  each,  spent   $1275  in 
improving  the  houses,  and  put  up  752 J  rods  of  fence  at  $  2}  per  rod. 
He  then  sold  the  farms  for  $  35|  per  acre.    Did  he  gain  or  lose,  and 
how  much  ? 

29.  A  and  B  joined  in  purchasing  a  farm  costing  $  4500,  A  pay- 
ing  $  2000  and  B  the  remainder.     After  owning  the  farm  six  months 
they  sold  it  for  $  6300.     Of  this  sum  how  much  should  each  receive  ? 

80.  By  what  number  must  $•  be  multiplied  to  produce  2401  ? 

81.  The  difference  between  \  and  £  of  a  number  is  90  less 
than  £  of  the  same  number.     What  is  the  number? 

82.  If  a  man  can  dig  20  bushels  of  potatoes  in  a  day,  and  can 
pick  up  30  bushels  in  a  day,  how  many  bushels  can  he  dig  and  pick 
up  in  20  days  ? 

83.  D  can  cut  f  of  a  cord  of  wood  in  |  day.     In  1  day  E  can  cut 
|  as  much  as  D  can  cut  in  a  whole  day.     If  they  work  together,  how 
long  will  it  take  them  to  cut  70  cords  ? 


92  FRACTIONS  [§273 

34.  If  3  J  acres  of  land  cost  $  65,  what  will  125^  acres  cost  at 
the  same  rate  ? 

35.  A  works  at  the  rate  of  $2f  a  day,  and  B  at  the  rate  of  $3£ 
a  day.    How  long  will  it  take  A  to  earn  as  much  as  B  earns  in 
19  days  ? 

86.  A  tank  has  an  inlet  by  which  it  may  be  filled  in  10  hours, 
and  an  outlet  by  which,  when  filled,  it  may  be  emptied  in  6  hours. 
If  both  inlet  and  outlet  be  opened  when  the  tank  is  full,  in  what 
time  will  it  be  emptied  ? 

87.  A  arid  B  are  engaged  to  perform  a  certain  piece  of  work  for 
$  35.55.     It  is  supposed  that  A  does  ^  more  work  than  B,  and  they 
are  to  be  paid  proportionately.     How  much  should  each  receive  ? 

88.  If  you  buy  60  lemons  at  the  rate  of  6  for  10^,  and  twice  as 
many  more  at  the  rate  of  5  for  8  ^,  and  sell  the  entire  lot  at  the  rate 
of  3  for  4^,  will  you  gain  or  lose,  and  how  much  ? 

89.  Henry  bought  a  basket  of  oranges  at  the  rate  of  3  for  2^, 
and  gained  50^  by  selling  them  at  the  rate  of  2  for  3^.     How  many 
oranges  did  he  buy  ? 

40.  There  are  108  bushels  of  corn  in  2  bins.    In  one  of  the  bins 
there  are  12  bushels  less  than  one  half  as  many  bushels  as  in  the 
other.     How  many  bushels  in  each  ? 

41.  Three  brothers  join  in  paying  off  the  mortgage  on  their 
father's  farm.     The  eldest  pays  -£  of  it,  and  the  others  pay  the 
remainder  in  equal  shares.     If  the  eldest  brother  pays  $90  more 
than  the  amount  paid  by  each  of  his  younger  brothers,  what  is  the 
amount  of  the  mortgage  ? 

42.  A  can  dig  1\  bushels  of  potatoes  in  \  of  a  day,  and  B  can 
dig  5J  bushels  in  \  of  a  day.    How  many  bushels  can  they  both  dig 
in  7f  days  ? 

43.  A  dealer  bought  250}  bushels  of  corn  at  60f  ^  per  bushel. 
If  he  sold  the  whole  amount  of  his  purchase  at  65^  per  bushel, 
what  was  his  gain  ? 

44-  Having  bought  120|  cords  of  wood  at  $  5 £  per  cord,  I  sold 
J  of  it  at  $  6  per  cord  and  the  remainder  for  $  340.  Did  I  gain  or 
lose,  and  how  much  ? 

45.  What  is  the  value  of  8  pieces  of  dress  silk  containing  481, 
422,  452,  401,  43s,  422,  452,  and  421  yards  at  $1£  per  yard  ? 


§§273-275]  DECIMAL  FRACTIONS  93 

46.  I  bought  240  J  bushels  of  oats  at  30}^  per  bushel,  190| 
bushels  of  corn  at  60 \$  per  bushel,  and  30  bushels  of  wheat  at 
$  1.12£,  and  sold  the  whole  for  $  320.     Did  I  gain  or  lose,  and  how 
much? 

47.  Find  the  total  cost  of  the  items  in  the  oral  exercise,  page  85; 
of  the  items  in  the  oral  exercise,  page  86. 

48.  A  man  bought  5  bags  of  wheat,  weighing  respectively  120, 
1L'4«,  128J,  132|,  and  131J  pounds,  at  $11  per  bushel.     If  each  bag, 
independent  of  the  wheat  it  contained,  weighed  1  pound,  and  there 
are  60  pounds  in  a  bushel  of  wheat,  did  he  gain  or  lose  by  selling 
the  whole  purchase  for  $  15  ? 

49.  A  produce  dealer's  sales  for  a  day  are  as  follows:   3411 
bushels  of  wheat  at  $11-  per  bushel,  410 \  bushels  of  barley  at  80^ 
per  bushel,  1120^  bushels  of  oats  at  30J^  per  bushel,  310^  bushels  of 
buckwheat  at  $  f  per  bushel,  250  bushels  of  beans  at  $  2-J-  per  bushel, 
13861  bushels  of  potatoes  at  501^  per  bushel,  1050J  bushels  apples 
at  $  i  per  bushel,  and  6301  bushels  of  turnips  at  70J  ^  per  bushel. 
Find  the  total  sales  for  the  day. 

50.  The  six  fields  of  a  farm  measure,  respectively,  10,  12J,  19}, 
26  T\,  301,  and  2-f^  acres,  and  are  valued  at  $  250  per  acre.    How 
much  is  the  farm  worth  ? 


DECIMAL  FRACTIONS 

274.   A  decimal  fraction,  or  a  decimal,  is  a  fraction  having  for  its 
denominator  10  or  some  power  of  10  ;  as,  - 


If  a  unit  be  divided  into  ten  equal  parts,  the  parts  are  called  tenths;  if  a 
tenth  of  a  unit  be  divided  into  ten  equal  parts,  the  parts  are  called  hundredths  ; 
if  a  hundredth  of  a  unit  be  divided  into  ten  equal  parts,  the  parts  are  called 
thousandths;  and  soon. 

To  obviate  the  trouble  of  writing  the  denominators  of  decimal  fractions,  an 
abbreviated  method  of  notation  is  used  as  shown  in  the  following  examples  : 


275.  Compared  with  Common  Fractions.  Decimal  fractions  are  in 
most  respects  quite  similar  to  common  fractions.  The  points  of  dif- 
ference may  be  stated  as  follows: 


94  TRACTIONS  [§§  275-281 

1.  The  denominator  of  a  common  fraction  is  always  written, 
while  that  of  a  decimal  fraction  is  only  indicated. 

2.  The  denominator  of  a  common  fraction  may  be  any  number, 
while  that  of  a  decimal  fraction  must  be  10  or  some  power  of  10. 

276.  The  decimal  point  is  a  period  (.).     It  is  always  placed  at 
the  left  of  tenths,  and  by  its  position  indicates  the  denominator  and 
determines  the  value  of  the  decimal  fraction ;  as,  .4,  .47,  .315. 

When  the  decimal  point  is  used  to  separate  the  integral  from  the  fractional 
part  in  a  mixed  decimal,  or  dollars  and  cents  in  a  decimal  currency,  it  is  called 
a  separatrix.  The  figures  written  at  the  right  of  the  decimal  point  constitute 
the  numerator  of  the  fraction,  and  the  number  of  figures  written  at  the  right 
of  the  decimal  point  indicates  the  power  of  10  which  constitutes  the  denomi- 
nator of  the  fraction. 

277.  A  pure  decimal  corresponds  to  a  proper  fraction,  the  value 
being  less  than  1 ;  as  .5,  .27,  .207,  .3241. 

278.  A  mixed  decimal  corresponds  to  an  improper  fraction,  the 
value  being  more  than  1 ;  as,  8.17,  17.8,  24.113. 

279.  A  complex  decimal  corresponds  to  a  complex  fraction,  and 

(OOJA 
T7m  )> 
10|  )0y 

100 

280.  General  Principles.     1.  Decimals  increase  in  value  from  right 
to  left,  and  decrease  in  value  from  left  to  right  in  a  tenfold  ratio. 

2.  A  decimal  should  contain  as  many  places  as  there  would  be 
ciphers  in  its  denominator  if  written,  the  decimal  point  representing 
the  unit  1  of  such  denominator. 

3.  The  value  of  any  decimal  fraction  depends  upon  its  distance 
from  the  decimal  point. 

4.  Prefixing  a  cipher  to  a  decimal  is  equivalent  to  dividing  it  by 
10. 

5.  Annexing  one  or  more  ciphers  to  a  decimal  does  not  alter  its 
value. 

NUMERATION  OF  DECIMAL  FRACTIONS 

281.  The  abbreviated  method  used  to  indicate  decimal  fractions 
is  nothing  more  than  an  extension  of  the  method  by  which  whole 
numbers  are  represented. 


§&  281-283]  DECIMAL  FRACTIONS  95 

The  relation  of  orders  in  a  mixed  decimal  fraction  is  clearly 
shown  by  the  following 

NUMERATION  TABLE 


H   n 


i 


127392416     •     72519275 
THE  INTEGRAL  PART  THE  FRACTIONAL  PART 

The  above  number  Is  read,  one  hundred  twenty-seven  million,  three  hundred 
ninety-two  thousand,  four  hundred  sixteen  and  seventy-two  million,  five  hundred 
nineteen  thousand,  two  hundred  seventy-five  hundred-millionths. 

282.  The  order  of  a  decimal  fraction  may  be  found  by  numerating 
either  from  right  to  left  or  from  left  to  right,  but  it  should  be 
remembered  that  the  decimal  point  stands  in  the  position  of  the 
unit  1  in  the  decimal  denominator.     The  order  of  a  decimal  may 
usually  be  determined  by  inspection  if  the  fact  to  be  drawn  from 
the  following  illustration  be  observed. 

If  .29  be  numerated  from  the  right  as  in  integers,  the  point  Is  in  the  hun- 
dreds' place,  hence  read  twenty-nine  hundredths.  In  .1137  the  point  is  in  tbe 
ten-thousands'  place,  hence  read  eleven  hundred  thirty-seven  ten-thousandths; 
in  .031631  the  point  is  in  the  millions'  place,  hence  read  thirty-one  thousand,  six 
hundred  thirty-one  millionths. 

283.  Hence  the  following  rule : 

Numerate  from  the  decimal  point  to  determine  the  de- 
nominator. 

Read  the  decimal  as  a  whole  number,  and  give  to  it  the 
denomination  of  the  right-hand  figure. 

In  reading  whole  numbers  never  read  and  between  periods  or  between  hun- 
dreds and  tens  and  units.  Thus,  in  reading  615,  say  six  hundred  fifteen,  and 
not  six  hundred  and  fifteen.  In  reading  mixed  decimals  always  connect  the 
Integral  and  fractional  parts  by  and;  as  2.5,  read  two  and  five  tenths;  17.016, 
read  seventeen  and  sixteen  thousandths. 


96  FRACTIONS  [§§  283~28f 

ORAL  EXERCISE 
Kead  the  following  decimals : 

1.  .297.  7.   .2.  IS.  .02.  19.  638.6|. 

2.  .1471.          £.    .20.  ^.  .002.  80.  341.131^. 

3.  .2442.          P.   .200.  16.  .0002.  £7.  801.00801. 

4.  .105.          m   .2000.  m  .00002.  22.  6000.58302. 

5.  .963.          11.   .214698.         77.  .000002.  #£.  9001.00901. 

6.  .56007.      ./£.   4003755.       18.  136.251.  jg£  3000.00030003. 

NOTATION  OF  DECIMAL  FRACTIONS 

284.  Example.     Write  as  a  decimal  thirty-four  hundredths. 

SOLUTION.  Observe  that  in  writing  thirty-four  hundredths  as  a  common 
fraction  the  mental  operation  is  as  follows :  After  writing  34,  the  numerator, 
the  question  is,  "34  what?"  The  answer  is  "34  hundredths,"  and  100  is 
written  below  as  a  denominator.  The  result  is  ^.  In  writing  the  decimal 
form  of  the  same  fraction  reason  in  practically  the  same  way.  Write  34,  the 
numerator,  and  make  it  hundredths  by  placing  before  34  a  decimal  point,  which 
represents  1  of  the  decimal  denominator.  The  result  is  .34.  Notice  that  34 
occupies  two  places  corresponding  to  the  two  ciphers  in  the  denominator. 

285.  Hence  the  following  rule : 

Write  the  decimal  the  same  as  a  whole  number,  prefixing 
ciphers  when  necessary  to  give  each  figure  its  true  local  value. 

Place  the  decimal  point  before  the  left-hand  figure  of  the 
decimal. 

WRITTEN  EXERCISE 

Express  by  figures  the  following  decimals : 
1.   Twenty-six  hundredths.  3.   Six  ten-thousandths. 

9.   Twenty-seven  hundredths.  4.  Four  hundredths. 

6.  Five  and  seven  tenths. 

6.  Five  hundred  and  five  hundredths. 

7.  Twenty-two  hundred-thousandths. 

,     8.  Five  thousand  and  five  thousandths. 

9.  One  million  and  one  millionth. 

10.  Five  hundred  thousandths. 

11.  Five  hundred-thousandths. 


§§285-288]  DECIMAL   FRACTIONS  97 

18.   Seventy-seven  tenths. 

IS.   Two  thousand  two  thousandths. 

14-    Two  thousand  and  two  thousandths. 

15.  Eleven  and  one  hundred  seven  millionths. 

16.  Eighty-three  and  five  hundred  four  ten-thousandths. 

17.  Seven   hundred  ten  and  two  hundred  forty-three  hundred- 
thousandths. 

18.  Fifty-four  million  fifty-four  thousand  fifty-four  and  fifty-four 
million  fifty-four  thousand  fifty-four  ten-billionths. 

19.  Write  the  following  as  decimal  fractions : 

Io0>         i  oooo>         -M  oao"> 


REDUCTION  OF  DECIMAL  FRACTIONS 

286.  To  reduce  a  decimal  to  a  common  fraction. 

287.  Examples.     1.   Keduce  .035  to  a  common  fraction. 

SOLUTION.     The  decimal  .035  is  read  thirty- 
.035  =  yjj-§7F  =  -2-Shj-.      five  thousandths,  which  as  a  common  fraction  is 

writtpn       3  5  35 —      7 

win/ben  iggfl-.       HT<nF  —  "ZWG' 

Hence,  .036  as  a  common  fraction  in  its  simplest  form  is  equal  to  j^y. 

2.   Keduce  .66|  to  a  common  fraction. 

.66£     "  SOLUTION.     .66$  is  a  complex  decimal.    In  the  form 

100  of  a  complex  fraction  it  is  equal  to  -— J.     Divide  the 

20  0 

?r~  _   goo 2      numerator  of  the  complex  fraction  by  the  denominator 

100"  and  the  result  is  f 

288.  Hence  the  rule: 

Omitting  the  decimal  point  write  the  proper  denominator 
and  reduce  the  fraction  to  its  simplest  form. 

ORAL  EXERCISE 

Keduce  to  equivalent  common  fractions  in  their  lowest  terms : 

1.  .25.      8.  .72.      5.  .125.      7.   .33£.        9.   .025.      11.  .121 

2.  .24.      4.   .75.      6.  .016.     8.  .44$.      10.  .250.      12.   .16}. 


4.  .4025. 

7.   .42504. 

10.   .24042. 

IS.  .008J. 

5.  .2244. 

8.  .28828. 

11.   .08004. 

14.   .326|. 

6.  .1878. 

9.   .425. 

12.  .1146. 

16.   .3131. 

£§§  288-292 
WRITTEN  EXERCISE 

Reduce  to  equivalent  common  fractions  in  their  lowest  terms : 

1.  .62. 

2.  .105. 
8.  .372. 

Reduce  to  ordinary  mixed  numbers : 

16.  5.16.        18.  11.75.      20.  16.162.       22.   65.132.       24.   15.016. 

17.  13.205.    19.  31.135.    21.  81.1888.     0&  35.0012.     &?.  28.44|. 

289.  To  reduce  a  common  fraction  to  a  decimal. 

290.  Example.     Reduce  £  to  an  equivalent  decimal. 

.8  SOLUTION.     |  is  equal  to  $  of  4  units.     4  units  is  equal  to  40 

tenths ;  |  of  40  tenths  (4.0)  =  .8.    Hence  the  rule : 


291.  Divide  the  numerator  by  the  denominator. 

If  the  denominator  of  a  common  fraction  reduced  to  its  lowest  terms  con- 
tains any  other  factor  than  2  or  5  (the  prime  factors  of  10),  the  quotient  of  the 
numerator  by  the  denominator  cannot  be  expressed  as  a  complete  decimal; 
or.!66+. 


WRITTEN  EXERCISE 

Reduce  to  equivalent  decimals : 

*•  A-  ±  &•  7.  -fh.  10.  H-  19. 

*•  &•        5.  H-        *•  «•          Jtf.  A-         ^  -B- 

All  ft       1 8  O          8  *p       81  /_£       47 

tt*  °*     38*  •*    T2TP  f*    M*  **•     Tfll* 

ADDITION  OP  DECIMAL  FRACTIONS 

292.   Example.    Add  .7,  2.43,  .865, 11.5, 113.2675,  and  200.00165. 

.7  SOLUTION.     By  the  decimal  system  numbers  increase  in 

2.43  value  from  right  to  left  in  a  tenfold  ratio,  and  the  decimal 

OGK  point  separates  the  integral  from  the  fractional  part.    Hence, 

,./£  to  add  decimals  the  points  should  be  placed  so  that  they  fall 

in  the  same  vertical  line.     The  units  of  the  same  order  will 

then  fall  in  the  same  column.    The  result  of  the  addition 

200.00165       may  then  be  obtained  in  the  same  manner  as  in  simple 

328.76415      numbers. 


§293]  DECIMAL   FRACTIONS  99 

293.    Therefore  the  following  rule : 

Write  the  decimals  so  that  the  points  will  fall  in  the 
same  vertical  line. 

Add  as  in  whole  numbers  and  place  the  point  in  the  sum 
directly  below  the  points  in  the  numbers  added. 

WRITTEN  EXERCISE 

1.   Add  3.04,  25.001,  .67,  .2146,  and  819.256. 

&   Add  30.1257,  605.2146,  1000.864532,  and  16.25694. 

8.   Add  896.111,  9530.216753,  1111.230094,  and  1100.960005. 

4.  Find  the  sum  of  seventeen  and  forty-six  ten-thousandths, 
eighty-three  and  one  thousand  four  millionths,  five  hundred  two  and 
seventy-five   hundred-thousandths,  one  million  six  and  six  million 
one  billionths. 

5.  Add  fifty-six  thousand  one  hundred  twelve  and  one  thousand 
twenty  millionths,  six  and  ninety-seven  million  five  billionths,  one 
thousand  five  hundred  seventy-nine  and  twenty-six  thousand  twenty- 
one  hundred-thousandths. 

6.  I  buy  10  bales  of  cloth   containing  32J,  41-^-,  39^,  46f, 
29.875,  38^,  43|,  31-^,  42f|,  and  40.635  yards,  respectively.     How 
many  yards  in  my  purchase  ? 

7.  A  man  cut  16.5  cords  of  wood  the  first  week,  15.76  cords  the 
second  week,  and  9f  cords  the  third  week.     How  many  cords  did  he 
cut  in  all  ? 

8.  How  many  thousand  feet  of  lumber  in  4  piles  measuring  as 
follows :  16,815,  28,185,  16,189,  75,141  feet,  respectively  ? 

9.  How  many  tons  of  coal  in  5  car  loads  weighing  respectively 
22.815  tons,  21.86  tons,  1|  tons,  19.998  tons,  and  18.125  tons  ? 

10.  A  lumber  dealer  bought  6  car  loads  of  lumber  as  follows : 
1  car  of  16.185  thousand  feet  at  a  cost  of  $178.12,  1  car  of  15.998 
thousand  feet  at  a  cost  of  $198.37|,  1  car  of  14.17  thousand  feet  at  a 
cost  of  $132.78,  1  car  of  19.175  thousand  feet  at  a  cost  of  $300.96^-, 
1  car  of  20.156  thousand  feet  at  a  cost  of  $  418.50,  and  1  car  of  15.500 
thousand  feet  at  a  cost  of  $  175.     Find  the  total  number  of  thousand 
feet  bought  and  the  total  cost  of  same. 

11.  Add  211,  542,  171,  303,  461,  612,  801,  393,  and  242. 

12.  Add  1211,  973,  462,  1113,  43,  712,  863,  501,  1033,  721,  713,  and  50. 


100  FRACTIONS  [§§  294-296 

SUBTRACTION  OF  DECIMAL  FRACTIONS 

294.  Examples.     1.   Find  the  difference  between  .127  and  .102. 
-^27  SOLUTION.     Both  decimals  are  of  the  same  order  ;  hence,  write 
-,  Q2         the  subtrahend  under  the  minuend  and  subtract  as  in  whole  num- 
bers keeping  the  points  in  the  minuend,  subtrahend,  and  remainder, 
respectively,  in  the  same  vertical  line. 

2.   From  .7  take  .37. 

jj  SOLUTION.    Annexing  ciphers  to  a  decimal  does  not  alter  its 

nrr  value  ;  hence,  consider  .7  as  .70  and  the  decimals  are  of  the  same 
'-—  order  and  may  be  subtracted  as  whole  numbers.  The  result  is 
•**  then  .33. 

8.  Find  the  difference  between  .73  and  2. 

2*  SOLUTION.     2  being  a  whole  number  must  be  greater  than  the 

.73        decimal  fraction  .73.     Consider  two  decimal  ciphers  as  annexed 

1.27        to  2  and  subtract  as  in  whole  numbers.     The  result  is  then  1.27. 

295.  Hence  the  following  rule : 

Write  the  terms  so  that  the  points  fall  in  the  same  vertical 
line  and  subtract  as  in  whole  numbers,  placing  the  point  in  the 
remainder  below  the  points  in  the  other  terms. 

WRITTEN  EXERCISE 

Find  the  difference  between : 

1.  .13823  and  .668.  7.  3491.5  and  4246.1005. 

2.  2  and  .72152.  8.  24.6852  and  25. 

3.  6.7584  and  1.^32.  9.  250.98754  and  386245. 

4.  2.3  and  .753452.  10.  .0001  and  1000.01. 
6.   .900  and  .09.  11.  3  and  .00015. 

6.   .002  and  .200.  J*.  259.00702  and  156.07. 


MULTIPLICATION  OF  DECIMAL  FRACTIONS 

296.   Examples.     1.   Find  the  product  of  .09  x  .5. 
QQ  SOLUTION.     .09  =  ^  and  .5  =  ^.    Applying  the  rule  for  the 

multiplication  of  common  fractions  -fa  x  &  =  T$fo  =  .045.     The 

^        figures  to  the  right  of  the  decimal  point  always  form  the  numerator 

.045        of  the  decimal  fraction  and  the  number  of  figures  to  the  right  of  the 


§  297]  DECIMAL   FRACTIONS  ;'  i   V  101 

decimal  point  indicate  the  power  of  10  which  forms  the*deno''iimatcr';'h"epfc^,J. 
in  performing  the  above  multiplication,  arrange  the  work  as  shown  in  the  mar- 
gin. 9  x  5  =  45,  or  the  numerator  of  the  product.  The  denominator  of  the 
multiplicand  is  the  second  power  of  ten,  hence,  contains  two  ciphers.  The 
denominator  of  the  multiplier  is  the  first  power  of  10,  hence,  contains  one 
cipher.  The  denominator  of  the  product  will  thus  contain  three  ciphers  (2  +  1) 
equal  to  three  decimal  places  ;  hence,  .09  x  .5  =  .045. 

2.   Find  the  product  of  2.05  x  .007. 

2  05  SOLUTION.     Multiply  as  in  Example  1.     Since  the  number  of 

figures  in  the  product  is  one  less  than  the  number  of  decimal  places 
in  the  multiplicand  and  multiplier,  supply  the  deficiency  by  prefix- 


.01435      ing  a  cipher?  as  shown  in  the  margin.     The  result  is  then  .01435. 

297.   From  the  foregoing  solutions  the  following  rule  is  derived : 

Multiply  as  in  whole  numbers. 

From  the  right  in  the  product  point  off  a  number  of  deci- 
mal places  equal  to  the  sum  of  the  number  of  decimal  places 
in  tJie  multiplicand  and  multiplier. 

WRITTEN  EXERCISE 

1.  .52  x. 25x0x95  =  ?  4.   25000  x  .000024  =  ? 

2.  1625.426  x  .0725  =  ?  5.   .009  x  .00001  x  10000  =  ? 
8.  2.5  x  .095  =  ?  6.   716.0025  x  10.1005=  ? 

7.  Multiply  three  hundred  forty-seven  ten-thousandths  by  fifty- 
two  thousandths. 

8.  I  sold  14.4  bales  of  cloth,  each  bale  containing  61.625  yards 
at  62-jy  per  yard.     How  much  did  I  receive  ? 

9.  From  10.85  acres  of  wheat  a  farmer  harvested  31.875  bushels 
per  acre  and  sold  his  crop  at  97 \$  per  bushel.     How  much  was 
received  for  the  crop  ? 

10.  A  man  having  650  acres  of  land  sold  .625  of  it.     What  is  the 
remainder  worth  at  $  60.95  per  acre  ? 

11.  Sold  Mohawk  Valley  Lumber  Co.  for  cash,  185.998  thousand 
feet  of  pine  lumber  at  $  18.87^-  per  thousand.     Find  the  amount  of 
the  invoice. 

12.  A  man's  income  is  $3500  a  year.      If  his  average  daily 
expenses  are  $  8.23,  how  much  can  he  save  in  a  year  ? 


102  FRACTIONS  [§§  297-301 

IS.  Bought  of- Field  Bros.  &  Co.  three  car  loads  of  lumber  at 
$17.87£  per  thousand  feet.  The  cars  contained  21.255  thousand 
feet,  22.275  thousand  feet,  and  16.250  thousand  feefc,  respectively. 
Find  the  amount  of  the  invoice. 

14.  For  constructing  a  house  and  barn,  I  bought  46.21  thousand 
feet  matched  pine  at  $  21  per  thousand,  13.516  thousand  feet  of  sid- 
ing at  $  28.50  per  thousand,  11.260  thousand  feet  chestnut  at  $  32 
per  thousand,  4.68  thousand  feet  black  walnut  at  $45  per  thousand, 
58.66  thousand  feet  hemlock  at  $  6.25  per  thousand,  13.7  thousand 
brick  at  $  5.50  per  thousand,  9.28  thousand  feet  cherry  at  $  86  per 
thousand,  and  33.725  thousand  feet  hemlock  timber  at  $  11  per  thou- 
sand. What  was  the  total  cost  ? 

SHORT  METHODS 

298.  Changing  the  position  of  the  decimal  point  in  any  number 
affects  the  local  value  of  each  figure  in  that  number. 

Thus,  if  .721  be  written  7.21,  7  tenths  are  made  7  units;  2  hundredths,  2 
tenths  ;  and  1  thousandth,  1  hundredth.  Hence, 

299.  To  multiply  a  decimal  by  1  followed  by  any  number  of  ciphers, 

Move  the  decimal  point  to  the  right  as  many  places  as  there  are 
ciphers  in  the  multiplier. 

300.  To  multiply  a  number  by  .1,  .01,  .001,  etc.,  is  equivalent  to 
dividing  by  10, 100, 1000,  etc.     Hence, 

301.  To  multiply  a  number  by  .1,  .01,  .001,  etc., 

Move  the  decimal  point  as  many  places  to  the  left  as  there  are  places 
in  the  multiplier. 

ORAL  EXERCISE 

By  inspection,  find  the  value  of : 

t.  75.42  x  .01.  6.  .151  x  3000.  11.  500  x  .025. 

2.   15.216  x  100.  7.  32  x  .00001.  12.  1201  x  .001. 

S.  952.1003  x  10,000.        8.   .0001  x  75.  IS.  17.02  x  .001. 

4.  10,000  x  .72142.  9.   16.295  x  .0001.  14.  157.271  x. 0001. 

5.  321.49  X  100.  10.  .15  X  400.  Id.  40.002  x  .001. 


§§302-304]  DECIMAL  FRACTIONS  103 

302.  8.5  X  8.5  =  8i  x  8^-.     Hence,  to  multiply  together  numbers 
ending  in  .5,  proceed  as  explained  in  261-265. 

303.  Examples.    1.   Find  the  value  of  8.5  acres  of  prairie  land  at 
$  8.50  per  acre. 

j>"  K  SOLUTION.    .5  x  .5  =  .25.    9  x  8  =  72.    If  1  acre  cost  $  8.50, 

;*  8.5  acres  will  cost  8.5  times  $  8.50,  or  $  72.26. 

72.25 

2.  Find  the  value  of  17.5  acres  of  prairie  land  at  $  4.50  per  acre. 

SOLUTION.    The  sum  of  the  numbers  to  the  left  of  the  deci- 
mals  is  odd  ;  hence,  the  decimal  fraction  in  the  product  is  .75(|). 
'  £(.5)  of  17  +4  +  17  x  4  =  78.5.     Rejecting  the  .5,  which  was 

78.75         included  in  the  .75  first  written,  the  result  is  $78.75. 

8.  Find  the  cost  of  15.5  dozen  men's  kid  gloves  at  $  5.50  per  dozen. 
15.5  SOLUTION.     The  sum  of  the  numbers  to  the  left  of  .5  is  even  ;" 

t$  $  hence,  the  decimal  fraction  in  the  product  is  .25(£).     |(.5)  of 

'  15  +  6  +  15  x  6  =  86.     Hence,  the  required  result  is  $  85.26. 


ORAL  EXERCISE 
Find  the  value  of: 

1.  21.5  A.  at  9  5.50.  7.  125  Ib.  at  35^*       13.  105  Ib.  at  45  £ 

2.  17.5  doz.  at  $  5.50.  8.  16.5  doz.  at  $  3.50.  14.  255  Ib.  at  45  4. 

3.  16.5  doz.  at  $4.50.  9.  15.5  Ib.  at  8j£        15.  115  Ib.  at  55  £ 

4.  25.5  yd.  at  $  4.50.  10.  12.5  yd.  at  7.5  £      16.  115  gal.  at  35  f. 
6.  19.5  yd.  at  $  2.50.  11.  115  Ib.  at  5.5  £        17.  205  gal.  at  45  £ 
6.  15.5  yd.  at  $  2.50.  12.  135  Ib.  at  35  f.         18.  215  gal.  at  55  £ 

DIVISION  OP  DECIMAL  FRACTIONS 

304.  Division  is  the  reverse  of  multiplication.  Since  in  multi- 
plication the  decimal  places  of  the  two  factors  are  added  to  deter- 
mine the  number  of  decimal  places  in  the  product,  in  division  the 
number  of  decimal  places  in  the  quotient  is  found  by  subtracting  the 
decimal  places,  if  any,  in  the  divisor  from  those  in  the  dividend. 

*  In  this  example  and  all  similar  cases,  perform  the  multiplication  just  as  if  the 
decimal  point  were  to  the  left  of  the  5's;  then,  in  the  product  point  off  as  many 
places  as  there  are  decimal  places  in  the  numbers  multiplied.  Thus,  in  this  problem 
say,  or  think:  odd  number,  therefore  write  75.  |  of  12  +  3  -f  12  X  3  =  43^.  Reject 
-  which  has  been  included  in  the  75  just  tvritten  and  the  product  is  4375.  Point  off 
two  places  and  the  result  is  $43.75. 


104 


FRACTIONS 


[§§  305-306 


305.   Example. 
.17).085(.5 


Divide  .085  by  .17. 


85 


SOLUTION.  The  divisor  has  two  decimal  places,  and  the 
dividend  has  three  decimal  places,  therefore  point  off  one 
(2—1)  place  from  the  right  in  the  quotient. 


306.     Hence  the  following  rule : 

When  needed,  annex  ciphers  to  the  dividend  to  make  its 
places  equal  in  number  to  those  of  the  divisor. 

Divide  as  in  whole  numbers,  and  from  tfo  right  of  the 
quotient  point  off  as  many  places  as  the  number  of  places  in 
the  dividend  exceeds  those  in  the  divisor. 

Do  not  commence  the  division  until  the  number  of  decimal  places  in  the 
dividend  is  at  least  equal  to  the  number  of  decimal  places  in  the  divisor.  Supply 
any  deficiency  in  the  dividend  by  annexing  ciphers. 

If  the  divisor  and  dividend  have  the  same  number  of  decimal  places,  the 
quotient  obtained  to  the  limit  of  the  dividend  as  given  will  be  a  whole  number. 

If  the  number  of  decimal  places  in  the  dividend  be  greater  than  the  number 
of  decimal  places  in  the  divisor,  point  off  from  the  right  of  the  quotient  the 
number  of  places  equal  to  such  excess,  prefixing  ciphers  to  the  quotient  if 
necessary. 

If  after  division  there  be  a  remainder,  ciphers  may  be  annexed  to  it  and  the 
division  continued  to  exactness  or  to  two  or  three  places  ordinarily  demanded 
in  business  computations.  Such  ciphers  should  be  considered  as  parts  of  the 
dividend. 

ORAL  EXERCISE 
By  inspection,  find  the  value  of: 

1.  1-1.        8.   .In-.Ol.       15.     .001-100.       22.       .022-110. 

2.  l-s-.l.        9.   .l-i-.OOl.     16.    .0001-*-.!.         23.         2.2-.00011. 

3.  l-i-.Ol.    10.    .1+10.        17.      100-.01.        24.      2200-.00011. 

4.  10 -.1.      11.   .1-100.      18.       10-10000.    25.       .022-11000. 

5.  10-.01.    12.    1-10.        19.      .22-11.         26.      2200-.000022. 

6.  .1-1.       18.    1  +  100.      20.      2.2-.011.       27.    .00001-10000. 

7.  .l-s-.l.      14.    1-1000.    21.     220-11000.    28.    10000 -.0001. 


§§  306-308] 


DECIMAL  FRACTIONS 


105 


WRITTEN  EXERCISE 
Find  the  sums  oi  the  quotients  in  the  following  problems : 

3. 

64-4-16. 
.64-?- 16. 
64  -*-  .16. 
640  ^  .16. 
640  -5-  1600. 
64 --.016. 
64  -K.  00016. 
6400  -*-  .16. 
6400  +  .00016. 
640  -s- 16000. 

39  -*- 130, 
3900  -4-  .13. 
390  -H  13000. 
3900  -*- 130000. 
.039  -*- 13. 
.0039  -^  130. 
.00039  -- 13000. 
.000039  -*- 13. 
.0000039  -5- 1.3. 
.039  +  .  013. 


1. 

0. 

9-1-9. 

75-*-  250. 

9-*-  .9. 

7.5  -?-  2500. 

9-5-  .09 

75  -s-  .25. 

9  -s-  .009. 

7500  -i-  .25. 

.009  -j-  9. 

.75  --  2500. 

900  -5-  900. 

750  -f-  25000. 

900  -T-  .09. 

"7500  -i-  .0025. 

9000  -5-  .0009. 

.075  -5-  .025. 

900  -?-  9000. 

750  -j-  .0025. 

.009  -*-  90000. 

75  -*-  .000025. 

* 

6. 

11  -*-  22. 

150  -*-.3. 

110  -*-  .22. 

150  -?-  .03. 

11  -5-  .022. 

150  -r-  3000. 

1100  -*-  .22. 

150  -5-  .003. 

11000  -*-  .022. 

1500-*-  .03. 

110  -*-  2200. 

15  -j-  30000. 

1100  -f-  22000. 

15  -*-  .0003. 

11  -*-  .000022. 

1500  -*-  .003. 

11000  -*-  .22. 

150  -*-  .00003. 

11  •*-  2200. 

15-*-  300. 

SHORT  METHODS 

307.  To  divide  a  decimal  by  1  followed  by  any  number  of  ciphers, 

Move  the  decimal  point  in  the  dividend  to  the  left  as  many  places 
as  there  are  ciphers  in  the  divisor. 

308.  To  divide  a  number  by  .01,  .001,  or  1  preceded  by  any  number 
of  decimal  ciphers, 

Move  the  decimal  point  in  the  dividend  as  many  places  to  the  right 
as  there  are  places  in  the  divisor. 


106  FRACTIONS  [§  308 

ORAL  EXERCISE 

1.  897-1-10.  6.  357.16 -j- 1000.  P.  .113-1- .001. 

2.  1.37-5-1000.  6.     14.27 -j- 100.  10.  .171 -*- .0001. 
8.  17.3 -f- 10.  7.         .82 -f- 100.  11.  .75-!- ,0001. 
4.  2.47-f-lOOO.  &       .075-^.01.  12.  13.54 -*- .001. 

ORAL  REVIEW 

1.  The  sum  of  two  numbers  is  .3.    The  smaller  number  is  .05. 
What  is  the  product  of  the  two  numbers  ? 

2.  If  .75  of  a  mill  is  worth  $  7500,  what  is  .5  of  it  worth  ? 

S.  If  .75  of  a  stock  of  goods  is  worth  $  225,  what  is  three  times 
the  stock  worth? 

4.  Five  times  a  certain  decimal  is  .4.     What  is  the  decimal  ? 

5.  Three  times  a  certain  decimal  is  .15.     What  is  twelve  times 
the  same  decimal  ? 

6.  How  many  thousandths  in  seven  units  ? 

7.  J-.5xi  =  ? 

5.  The  product  of  two  numbers  is  .0006.  If  one  of  the  num- 
bers is  .03,  what  is  the  other  ? 

9.  The  sum  of  two  numbers  is  15.  If  one  of  them  is  6.5,  what 
is  the  product  of  the  numbers  ? 

10.  $2.50  is  how  many  hundredths  times  $75? 

11.  What  will  7.5  thousand  envelopes  cost  at  $  2.50  per  thousand  ? 

12.  Find  the  cost  of  12.5  thousand  feet  of  plank  at  $8.50  per 
thousand. 

WRITTEN  REVIEW 

•1.  The  sum  of  three  numbers  is  4.5.  If  the  smaller  is  .95  and 
the  larger  2.05,  what  is  the  product  of  the  three  numbers  ? 

2.  Multiply  the  sum  of  sixty-five  hundred  and  sixty-five  and  one 
hundred  seven  millionths  by  the  product  of  nine  hundred  millionths 
and  one  hundred  twenty-seven  and  seventeen  hundredths. 

8.  What  is  the  cost  of  6  barrels  of  sugar  weighing  301, 314, 297, 
309,  313,  and  315  pounds,  respectively,  at  6J^  per  pound? 


§  308]  DECIMAL  FRACTIONS  107 

4.  If  a  wheelman  travels  10.3  hours  per  day,  how  many  days 
will  be  required  for  him  to  travel  558.0025  miles  at  the  rate  of  7.88 
per  hour  ? 

5.  I  sold  a  lumber  man  381.25  pounds  of  butter  at  28|  ^  per 
pound,  2468.375  pounds  of  cheese  at  11.4^  per  pound,  and  2356.5 
pounds  of  dressed  beef  at  7f  ^  per  pound,  and  received  pay  in 
lumber  at  $  23.12 £  per  thousand  feet.     How  many  thousand  feet  of 
lumber  should  I  have  received  ? 

6.  A  man's  salary  is  $2500  per  year.     If  he  spends  $650.25 
for  board,  $119.25  for  books,  $31.85  for  other  literature,  $63.40  for 
charity,  $209.75  for  clothes,  $  109.90  for  traveling  expenses,  $115.60 
for  incidental  expenses,  and  saves  the  remainder,  how  long  will 
it  take  him  to  pay  for  a  piece  of  property  valued  at  $  8400  ? 

7.  A  merchant  had  on  hand  Jan.  1, 1904,  a  stock  of  merchan- 
dise  aggregating  $11750.90.     During  the  year  he  bought  goods 
amounting  to  $7315.90  and  sold  goods  amounting  to  $15364.85. 
If  on  Dec.  31,  1904,  he  has  stock  on  hand  valued  at  $9215.75,  has 
he  gained  or  lost  for  the  year  and  how  much  ? 

8.  Having  bought  25  gross  of  steel  pens  at  $1.25  per  gross, 
I  sold  them  at  12^  each.    If  there  are  144  pens  in  a  gross,  did  I 
gain  or  lose,  and  how  much  ? 

9.  Find  the  total  cost  of  the  items  in  the  oral  exercise,  page  103. 

10.  If  a  boy  receives  $1.25  a  day,  and  a  man  $3.75  a  day,  how 
long  will  it  take  the  boy  to  earn  as  much  as  the  man  can  earn  in 
16  days  ? 

11.  In  a  certain  business  school  .5  of  the  students  study  book- 
keeping, .75  of  the  remainder  study  shorthand  and  typewriting,  and 
the  remainder,  125  pupils,  study  the  English  branches.     How  many 
students  in  each  department,  and  in  the  entire  school  ? 

12.  C.  W.  Allen  bought  of  J.  E.  Seel  &  Co.,  dealers  in  flour  and 
feed,  135  barrels  roller  process  flour  at  $6.75  per  barrel,  135  barrels 
searchlight  pastry  flour  at  $  5.75  per  barrel,  375  sacks  puritan  pan- 
cake flour  at  23  ^  per  sack,  195  sacks  chef  pastry  flour  at  25  ^  per 
sack,  250  bags  bran  at  $1.50  per  bag,  1500  pounds  corn  meal  at 
2J^  per  pound.     Find  the  amount  of  the  bill. 


108  FRACTIONS  [§§  308-315 

IS.  A  and  B  are  in  partnership.  A  is  to  receive  .75  of  the 
profits  and  B  the  remainder.  At  the  end  of  one  year  B  draws 
$  1250  as  his  share  of  the  profits.  If  the  total  losses  for  the  year 
were  $  950,  what  was  the  total  gain  for  the  year  ?  the  net  gain  ? 

14>  January  1,  A  and  B  join  in  the  purchase  of  some  real  estate, 
A  paying  .4  of  the  purchase  price,  and  B  the  remainder.  They  share 
the  profits  arising  from  the  sale  in  proportion  to  their  investments. 
The  property  is  sold  at  a  profit  of  $  2500  and  B  receives  $  6000  as 
his  share.  How  much  did  A  and  B  pay  for  the  real  estate  ? 

QUANTITY,  PRICE,  AND  COST 

309.  The  essential  elements  of  every  business  transaction  in- 
volving the  money  value  of  property  or  labor  are  quantity,  price,  and 
cost. 

310.  The  fixed  unit  used  in  estimating  the  money  value  of  com- 
modities is  termed  a  commercial  unit. 

A  yard,  a  dozen,  a  bushel,  and  an  acre  are  commercial  units. 

311.  Quantity  is  the  number  of  commercial  units  in  any  given 
commodity. 

312.  Price,  is  the  value  put  upon  a  commercial  unit. 

313.  Cost  is  the  value  of  a  quantity. 

314.  An  aliquot  part  of  a  number  is  one  of  the  even  parts  of  that 
number. 

20,  25,  33$,  50,  etc.,  are  aliquot  parts  of  100. 

315.  The  unit  of  an  aliquot  part  is  the  number  which  must  be 
divided  to  obtain  the  part. 

$1  is  the  unit  of  the  aliquot  parts  20  ?,  25  ?,  83Jft  etc. 


DRILL  EXERCISE 

1.  Name  three  commodities  of  which  a  dozen  is  the  commercial 
unit  ;  a  yard  ;  a  pound  ;  a  ton. 

2.  Name  three  quantities  of  which  the  commercial  unit  is  1  rod  ; 
1  acre  ;  1  barrel  ;  1  bag  ;   1  bale. 

8.  Name  three  aliquot  parts  of  1  yard;  of  1  bushel;  of  1  day; 
of  ft 


314-317] 


QUANTITY,   PKICE,    AND   COST 


109 


4.   Name  three  aliquot  parts  of  50  ;  of  250  ;  of  J. 

6-   Name  four  aliquot  parts  of  1  ton  ;  of  2  yards  ;  of  30. 

6    What  aliquot  part  of  $1  is  50^?     33^?     25^? 


7.  What  aliquot  part  of  25^  is  12^? 
?    1|^? 

8.  What  aliquot  part  of  50^  is  2^? 


316.   The  aliquot  parts  of  $  1  are  especially  useful  in  computa- 
tions where  the  quantity  and  price  are  given  to  find  the  cost. 

TABLE  OP  ALIQUOT  PARTS 


PARTS  OF  $1.00 


PABTS  OF  50  j? 


PARTS  OF  25  p 


PARTS  MORB  OR  LESS  THAN  $  1.00 


112|  ?  =  $  more 


133J  $  =  \  more 
110  ^   =  ^  more 


83^  =  i  less 


=  f  less 
40  ^  =  f  less 
37^  =  f  less 


317.   Many  of  the  ordinary  business  computations  may  be  mate- 
rially shortened  by  the  use  of  aliquot  parts. 


DRILL  EXERCISE 

1.  Formulate  a  short  method  for  finding  the  cost  of  a  quantity 
when  the  price  is  33£  ^. 

SOLUTION.    Since  33^  is  £  of  $1,  to  find  the  cost  of  a  quantity  when  the 
price  is  33£  ^  consider  the  quantity  as  dollars  and  divide  by  3. 

2.  Formulate  a  short  method  for  finding  the  cost  of  a  quantity 
when  the  price  is  50^;  25^;  20/;  16f^;  12J^;  6J^;  8J  £ 


110 


FRACTIONS 


[§§  317-319 


8.  Formulate  a  short  method  for  finding  the  cost  of  a  quantity 
(a)  when  the  price  is  10  $  ;  (6)  when  the  price  is  5  £ 

SOLUTIONS,  (a)  Since  10  p  is  fo  of  $  1,  to  find  the  cost  of  a  quantity  when 
the  price  is  Wp,  point  off  from  the  right  one  place  in  the  quantity  considered  as 
dollars. 

(6)  Since  5j*  is  %  of  10^,  to  find  the  cost  of  a  quantity  when  the  price  is  5^, 
point  off  one  place  in  the  quantity  considered  as  dollars  and  divide  by  2. 

4.  Formulate  a  short  method  for  finding  the  cost  of  a  quantity 
when  the  price  is  3^;  2J^;  lf^;  1}^;  lj£ 

318.  GENERAL  RULE,  find  the  cost  of  the  total  quantity 
by  multiplying  $  1  by  the  given  quantity;  then  take  such 
part  of  the  product  thus  obtained  as  the  given  price  is  a  part 


319.  An  abbreviation  is  a  part  of  a  word  used  to  indicate  an 
entire  word. 

Many  abbreviations  are  used  in  computations  involving  quantity, 
price,  and  cost.  The  most  important  of  these  are  shown  in  the 
following  list 

BUSINESS  ABBREVIATIONS 


Al.    . 

first  quality 

doz.   .    . 

dozen 

No.  .  nnmber 

Apr.    . 

April 

Dr.    .    . 

debtor 

Nov.   November 

acct.    . 

account 

ea.    .    . 

each 

Oct..  October 

amt.    . 

amount 

E.&O.E. 

errors  and  omis- 

oz.   .  ounce 

Aug.    . 

August 

"sions  excepted 

p.     .  page 

bal.    . 

balance 

Feb.  .    . 

February 

pp.  .  pages 

bbl.      . 

barrel 

ft..    .    . 

foot  or  feet 

pt.    .  pint 

B/L    . 

bill  of  lading 

f  rt.     .    . 

freight 

payt.  payment 

bot.  .    . 

bought 

gaL    .\ 

gallon 

Pd.  .  paid 

bu.  .    . 

bushel 

gro.   .    . 

gross 

pkg..  package 

bx.  .    . 

box 

hhd.  .    . 

hogshead 

pc.    .  piece 

cd.  .    . 

cord 

hr.     .    . 

hour 

pr.    .  pair 

ctg.  .    . 

cartage 

in.      .    . 

inch 

qt.    .  quart 

Co..    . 

Company 

Jan.  .    . 

January 

reed,    received 

C.O.D. 

collect  on  delivery 

Jr.     .    . 

Junior 

R.R..  railroad 

Cr.  .    . 

creditor 

Ib.     .    . 

pound 

sec.  .  second 

cwt.     . 

hundredweight 

Mar.  .    . 

March 

s.  .    .  shilling 

da.  .    . 

day 

mem.  .    . 

memorandum 

Sept.    September 

d.    .    . 

pence 

mo.    .    . 

month 

Sr.    .  Senior 

Dec.     . 

December 

Messrs.  . 

Gentlemen  or  Sirs 

wk.  .  week 

disc.     . 

discount 

Mr.    .    . 

Mister 

yd.    .  yard 

do.  .    . 

ditto,  or  the  same 

Mrs.  .    . 

Mistress 

yr.    .  year 

31 9J 


QUANTITY,  PRICE,  AND  COST 
BUSINESS  CHARACTERS 


&  number 

@  at 

a/c  account 

c/o  care  of 

x  by  (in  surface 

8  dollars 


1* 

ia 
1» 


cents 

1* 

If  or  11 

If 

per  cent 

by  the  thousand 


C  by  the  hundred 

^  check  mark 

**  ditto 

'  feet 

"  inches 

£  pounds  sterling 


ORAL  EXERCISE 


By  inspection,  find  the  cost  of: 

1.  350  Ib.  tea  at  50  £ 

2.  870  Ib.  coffee  at  33^. 

3.  124  Ib.  raisins  at  25  £ 

4.  24  Ib.  raisins  at  16f  £ 

5.  190  Ib.  rice  at  10  £ 

6.  160  Ib.  seed  at 

7.  123  Ib.  meal  at 

8.  855yd.  prints  at  20  £ 

9.  144  yd.  gingham  at 

10.  180  yd.  silesia  at  16f  £ 

71.  192  yd.  lining  at 

12.  1140  yd.  prints  at 

18.  284  yd.  lining  at 

1J.  960  yd.  ticking  at 


15.  168  Ib.  hamatl6f£ 

70.  368  yd.  plaids  at 

27.  88  yd.  lace  at  87|  £ 

18.  340  yd.  mohair  at  75  £ 

19.  390  yd.  alpaca  at  66f  £ 

20.  484  Ib.  lard  at  12}  £ 

21.  1680  doz.  eggs  at  16f  f. 

22.  240  Ib.  pork  at 

28.  1152  yd.  linen  at 
«£  728  gal.  cider  at 
#5.  Ill  qt.  berries  at 
#0.  880  Ib.  salt  at  lj 
#7.  164  yd.  cotton  at 
£<?.  86  Ib.  meal  at 


DRILL  EXERCISE 

1.  Formulate  a  short  method  for  finding  the  quantity  when  the 
cost  is  given  and  the  price  is  33^. 

SOLUTION.  Since  the  price  is  contained  3  times  in  $1,  the  quantity  bought 
will  be  3  times  the  number  of  dollars  invested ;  hence,  multiply  the  cost  by  S 
and  consider  the  result  as  quantity. 

2.  Formulate  a  short  method  for  finding  the  quantity  when  the 
cost  is  given  and  the  price  is  50  ^j  25^;  20  ^j  16f  ^ 


112  FRACTIONS  [§  319 

3.  Formulate  a  short  method  for  finding  the  quantity  (a)  when 
the  cost  is  given  and  the  price  is  10^;  (b)  when  the  price  is  3^. 

SOLUTIONS,  (a)  Since  10^  is  contained  10  times  in  $  1,  when  the  cost  is 
given  and  the  price  is  10^,  annex  a  cipher  to  the  dollars  and  consider  the  result 
as  quantity. 

(6)  10  f  is  3  times  31^.  If,  to  find  the  quantity  at  10^,  we  annex  a  cipher 
to  the  cost  and  consider  the  result  as  quantity,  to  find  the  cost  at  3^,  we  should 
annex  one  cipher  to  the  cost,  multiply  by  5,  and  consider  the  result  as  quantity. 

4.  Formulate  a  short  method  for  finding  the  quantity  when  the 
cost  is  given  and  the  price  is  21^;  lf^ 


ORAL  EXERCISE 

1.  How  many  pounds  of  tea  worth  33^  can  be  bought  for 
$123? 

2.  How  many  pounds   of   tea  worth  50^  can   be   bought   for 
$419.50? 

8.   How  many  yards  of  cloth  at  16f  /  can  be  bought  for  $25  ? 

4.  At  25^,  how  many  yards  of  prints  can  be  bought  for  $25  ? 

5.  At  33J^,  how  many  gallons  of  molasses  can  be  bought  for 
$15? 

6.  At  3J^,  how  many  pounds  of  dried  apples  can  be  bought 
for  $32? 

7.  At  21^,  how  many  pounds  of  sugar  can  be  bought  for  $  12? 

8.  At  3J  #,  how  many  pounds  of  sugar  can  be  bought  for  $  18  ? 

9.  At  20^,  how  many  yards  of  cotton  can  be  bought  for  $  125  ? 

10.  How  many  yards  of  gingham  at  8^  can  be  bought  for  $  11  ? 

11.  How  many  yards  of  ticking  at  6|^  can  be  bought  for  $  8  ? 

12.  How  many  yards  of  lining  worth  6|  ^  can  be  bought  for  $  15  ? 


WRITTEN  EXERCISE 

1.  A  farmer  sold  26£  bu.  buckwheat,  at  87£  f  per  bu.,  and  took 
his  pay  in  sugar  at  6J^  per  Ib.     How  many  pounds  should  he  have 
received  ? 

2.  A  gardener  exchanged  132  qt.  of  berries,  at  8J^  per  qt.,  and 
75  doz.  corn,  at  12£  $  per  doz.,  for  cloth  at  25^  per  yd.     How  many 
yards  did  he  receive  ? 


§§  319-321J  QUANTITY,    PRICE,   AND   COST  113 

8.  If  I  exchange  1920  acres  of  wild  land,  at  $  7.50  per  acre,  for 
an  improved  farm  at  $  125  per  acre,  what  should  be  the  number  of 
acres  in  my  farm  ? 

4.  A  farmer  gave  8|  cwt.  of  pork,  at  $  7.50  per  cwt.,  15  bu.  of 
beans,  at  $  3.25  per  bu.,  and  46£  bu.  of  oats,  at  33^  per  bu.,  for  28 
yd.  of  dress  silk,  at  $  1.25  per  yd.,  and  52J  yd.  of  delaine,  at  16|  $ 
per  yd.,  receiving  for  the  remainder,  cotton  goods  at  12£^  per  yd. 
How  many  yards  of  cotton  goods  should  be  delivered  to  him  ? 

5.  When  potatoes  are  worth  66|  ^  per  bu.,  and  turnips  25  ^  per 
bu.,  how  many  pounds  of  coffee,  at  16f  ^  per  lb.,  will  pay  for  24  bu. 
of  potatoes  and  18  bu.  of  turnips  ? 

6.  Having  bought  1487  lb.  A.  sugar,  at  6J^  per  lb.;  872  lb.  C. 
sugar,  at  5^  per  lb.;  628|  lb.  Y.H.  tea,  at  33^  per  lb. ;  522  lb.  J. 
tea.,  at  25^  per  lb. ;  650  lb.  Rio  coffee,  at  12£^  per  lb. ;  and  81  sacks 
of  flour,  at  $1.25  per  sack,  I  give  in  payment  seven  one-hundred 
dollar  bills.    How  much  should  be  returned  to  me? 

GENERAL  APPLICATIONS  OF  ALIQUOT  PARTS 

320.  The  principles  of  aliquot  parts  may  be  applied  to  a  great 
many  business  exercises,  as  will  be  shown  in  the  following  examples. 

321.  Examples.    1.  What  will  be  the  cost  of  15  bbl.  of  pork  at 
$16f  per  barrel? 

SOLUTION.  Since  $16f  Is  j  of  $ 100,  16$  times  any  Dumber  is  |  of  100  times 
that  number.  Hence, 

To  multiply  by  16f ,  annex  two  ciphers  to  the  multiplicand  and  divide  the 
result  by  £,  obtaining  as  a  result  $250. 

8.  What  will  25  acres  of  land  cost  at  $  164  per  acre  ? 

SOLUTION.    25  acres  at  $  164  is  equal  to  164  acres  at  $  25  per  acre.    Hence, 

Annex  two  ciphers  to  164  and  divide  by  4,  obtaining  as  a  result  $4100. 

S.  Find  the  cost  of  25  yd.  of  cloth  at  44  £  per  yard. 
SOLUTION.     Interchange  25  and  44  and  apply  the  principles  of  aliquot  parts. 
Then,  44  -*•  4  =  11.    Therefore,  25  yd.  of  cloth  at  44?  will  cost  $  11. 

4.  Find  the  cost  of  2500  lb.  of  coffee  at  32  f  per  pound. 

SOLUTION.  2600  lb.  at  32^  per  pound  is  equal  to  3200  lb.  at  25  J*  per  pound. 
Hence, 

interchange  the  significant  figures  (in  this  case  32  and  25)  and  apply  the 
principles  of  aliquot  parts.  Then,  8200  +  4  =  800.  Therefore,  2600  lb.  of  coffee 
at  32?  will  cost  $800. 


114  FRACTIONS  [§  321 

5.  Find  the  cost  of  250  Ib.  of  tea  at  64  ^  per  pound. 

SOLUTION.  250  Ib.  at  64^  per  pound  is  equal  to  640  Ib.  at  25^  per  pound. 
Hence, 

Interchange  the  significant  figures  and  apply  the  principles  of  aliquot  parts. 
Then,  640  -j-  4  =  160.  Therefore,  250  Ib.  of  tea  at  64  p  will  cost  $  160. 

6.  Find  the  cost  of  250  acres  of  land  at  $  44  per  acre. 

SOLUTION.  250  acres  at  $  44  per  acre  is  equal  to  440  acres  at  $  25  per  acre. 
Annexing  two  ciphers  to  440  the  result  is  44,000.  Then,  44,000  -*•  4  =  11,000. 
Therefore,  250  acres  of  land  at  $44  per  acre  will  cost  $  11,000. 

DRILL  EXERCISE 

1.  Formulate  a  short  method  for  multiplying  any  number  by  12J. 

SOLUTION.  Since  12 £  is  £  of  100, 12£  times  any  number  is  |  of  100  times  the 
game  number.  Hence, 

To  multiply  any  number  by  12J,  annex  two  ciphers  to  the  multiplicand  and 
divide  by  8. 

2.  Formulate  a  short  method  for  multiplying  any  number  by  25 ; 
by  2.5;  by  3331;  by  1J;  by  125;  by  250;  by  66|;  by  6|;  by  33J; 
by  1250. 

3.  Demonstrate  that  2500  Ib.  of  coffee  at  44  f  per  pound  is  equal 
to  4400  Ib.  at  25^  per  pound. 

ORAL  EXERCISE 
By  inspection,  find  the  value  of: 

1.  72  head  cattle  at  $  25.  IS.  250  yd.  wool  crepon  at  $  2.40 

2.  75  acres  land  at  $33J.  14.  16|  gro.  buttons  at  24^. 

3.  48  bbl.  beef  at  9 16|.  16.  125  yd.  cheviot  at  $1.05. 

4.  162  bbl.  pork  at  $12J.  16.  250  yd.  taffeta  silk  at  88^. 

5.  25  Ib.  tea  at  640.  17.  45  gro.  buttons  at  33^. 

6.  250  acres  land  at  $44.  18.  250  acres  land  at  $  88. 

7.  2500  acres  land  at  $16.  19.  33£  Ib.  tea  at  54  <f. 

8.  45  bbl.  beef  at  $16£  20.  96  yd.  cloth  at  33j£ 

9.  125  yd.  cashmere  at  64^.  21.  24  tons  coal  at  $  12J. 

10.  333J  yd.  silk  at  96  £  22.  36  tons  coal  at  $  8J. 

11.  166 J  yd.  cotton  at  36  £  28.   32  tons  coal  at  $  6J. 

12.  64  acres  land  at  $  25.  24.   18  tons  coal  at  $3J. 


§321] 


QUANTITY,   PRICE,   AND   COST 


115 


WRITTEN  EXERCISE 
1.   Find  the  total  cost  of : 


236  Ib.  tea  at  50  £ 
1152yd.  linen  at 

528  Ib.  lard  at 

488  gal.  molasses  at  25^. 
1848  doz.  eggs  at  16f  £ 

2.   Find  the  total  cost  of  : 


350  yd.  plaids  at 
248  yd.  ticking  at  10  £ 
1140  yd.  prints  at  33  j£ 
950  yd.  lining  at  25  f. 
720  yd.  drilling  at 


5.   Find  the  total  cost  of: 

188  Ib.  of  ham  at  16f  £ 
250  Ib.  Japan  tea  at  44^. 
125  gal.  molasses  at  91^. 
25  sacks  flour  at  $  1.24. 
125  sacks  flour  at  $  1.80. 


844  yd.  cambric  at  25  £ 
250  yd.  alpaca  at  72  £ 
112yd.  silk  at  $2.50. 
3608  yd.  gingham  at  12^. 
25  yd.  brilliantine  at  $1.84. 


2500yd.  lining  at  41  £ 

250  yd.  gingham  at 
2500  yd.  mohair  at  $1.23. 

250  yd.  sateen  at  32^. 

125  yd.  diagonals  at  $  1.04. 


25  doz.  cans  of  corn  at  $  1.25. 
166|lb.  tea  at  42^. 
125  doz.  cans  peaches  at  $  1.28. 
250  cans  tomatoes  at  88^. 
125  doz.  cans  pears  at  $  1 .20. 


DRILL  EXERCISE 

.  1.   Formulate  a  short  method  for  finding  the  cost  when  the  quan- 
tity is  given  and  the  price  is  $  1.66J . 

SOLUTION.  $1.66f  is  10  times  16§^  ;  hence,  to  find  the  cost  when  the  price  is 
$1.66|,  annex  a  cipher  to  the  quantity  considered  as  dollars  and  divide  by  6. 

2.  Formulate  a  short  method  for  finding  the  cost  when  the  quan- 
tity is  given  and  the  price  is  $3.33£;  $1.25. 

3.  How  may  the  cost  be  found  when  the  quantity  is  given  and 
the  price  is  66J^? 

SOLUTION.  66 f  is  10  times  6f .  6f  is  ^  of  a  dollar  ;  hence,  to  find  the  cost 
when  the  price  is  66f  ^,  add  a  cipher  to  the  quantity  considered  as  dollars  and 
divide  by  15. 


116  FRACTIONS  [§  321 

4.  Formulate  a  short  method  for  finding  the  cost  when  the 
quantity  is  given  and  the  price  is  $6.66}. 

5.  How  may  the  cost  be  found  (a)  when  the  price  is  75^? 
(6)  when  the  price  is  $1.33J? 

SOLUTIONS,  (a)  75f  is  \  less  than  $1;  hence,  to  find  the  cost  when  the 
price  is  75^,  consider  the  quantity  as  dollars  and  subtract  -.  of  itself . 

(6)  #1.33$  is  I  more  than  $1 ;  hence,  to  find  the  cost  when  the  price  is 
$1.33|,  to  the  quotient  considered  as  dollars  add  J  of  itself. 

6.  Formulate  a  short  method  for  finding  the  cost  when  the 
quantity  is  given  and  the  price  is  $7.50;  $75;  $1.25;  $1.08}. 

7.  Formulate  a  short  method  for  finding  the  quantity  when  the 
cost  is  given  and  the  price  is  $  1.66}. 

SOLUTION.  When  the  price  is  16|^,  the  cost  is  \  of  the  quantity.  When 
the  price  is  $1.66$,  the  cost  is  10  times  £  of  the  quantity;  hence,  to  find  the 
quantity  when  the  price  is  $1.66$,  point  off  one  place  in  the  dollars  considered 
a's  quantity  and  multiply  by  6. 

.  8.   Formulate  a  short  method  for  finding  the  quantity  when  the 
cost  is  given  and  the  price  is  $6.66};  $2.50. 

9.   How  may  the  quantity  be  found  when  the  cost  is  given  and 
the  price  is  $  7.50  ? 

SOLUTION.  75^  plus  $  of  itself  is  equal  to  $1 ;  henco,  when  the  price  Is  75^, 
I  of  the  cost  added  to  itself  is  equal  to  the  quantity.  $7.50  is  10  times  75^; 
hence,  to  find  the  quantity  when  the  price  is  $  7.50,  point  off  one  place  in  the  cost 
considered  as  quantity  and  add  J 

10.  Formulate  a  short  method  for  finding  the  cost  when  the 
quantity  is  given  and  the  price  is  $25;  $75. 


ORAL  EXERCISE 
Find  the  cost  of : 

1.  240  Ib.  at  75  £  6.  750  Ib.  at  84^.  11.  300  Ib.  at  42  £ 

0.   165  Ib.  at  $3.33}.  7.  1000  Ib.  at  1\$.  12.  3000  Ib.  at  16 £ 

3.  366  Ib.  at  $  1.66}.  8.  125  Ib.  at  88  f.  18.  2500  Ib.  at  $  .444 

4.  333}  Ib.  at  24  £  9.  484  Ib.  at  $  2.50.  14.  250  Ib.  at  16  £ 

5.  166}  Ib.  at  66  £  10.  33}  Ib.  at  99^.  15.  125  Ib.  at  64  £ 


QUANTITY,  PRICE,  AND  COST 


117 


WRITTEN  EXERCISE 

In  the  following  problems  make  all  extensions  mentally. 
1.   Find  the  total  cost  of: 


316  Ib.  at  10  £ 
484  Ib.  at  250. 
1000  Ib.  at  71  £ 
2500  Ib.  at  16  £ 
3000  Ib.  at  11  £ 

216  Ib.  at  12i  £ 
1124  Ib.  at  50  £ 
320  Ib.  at  61  £ 
816  Ib.  at  25  £ 
381  Ib.  at  3340. 

1095  Ib.  at  33£j 
125  Ib.  at  640. 
95  Ib.  at  81  £ 

1445  Ib.  at  20  £ 

2.  Find  the  total  cost  of . 

835  yd.  at  10  0.  298  yd.  at  50  0. 

450  yd.  at  33J0.  333  yd.  at  33J0. 

288  yd.  at  26  £  240  yd.  at  8J0. 

250  yd.  at  44  £  966  yd.  at  16f  0. 

t200yd.at  $2.50.  750  yd.  at  6f  £ 

5    Find  the  total  cost  of : 

1400  Ib..  at 
2163  Ib.  at 
7200  Ib.  at  50. 
1250  Ib.  at  8.4^. 
125  Ib.  at  6.4  £ 

4.  Find  the  total  cost  of? 
525  yd.  at  20  £  360  yd.  at  750. 


3980  Ib.  £ 
128  Ib.  at  16f  £ 
291  Ib.  at  500. 

1437  Ib.  at  250. 
840  Ib.  at 


28533yd.at  100. 
1400  yd.  at  610. 

450  yd.  at  6f  0. 

364yd.  at  $1.25. 


280yd.  at  $2.50. 
320  yd.  at  75  £ 
146  yd.  at  25?'. 
2500yd.  at  88  £ 


5.  Find  the  total  cost  of  : 


1095  yd.  at  3^. 

484  yd.  at  6  J  £ 

366  yd.  at  8^  £ 

1291  yd.  at  11  £ 

2502  yd.  at  24  f. 


400  yd.  at 
756  yd.  at  16|  £ 
320  yd.  at  12|  £ 
1515  yd.  at  66f  £ 
906  yd.  at 


1000  yd.  at 

2000yd.  at  21  £ 
648yd.  at  12£* 
125yd.  at  88  £ 

2500  yd.  at  64  £ 


880  Ib. 
832lb.  at  50^ 
1072lb.  at  50^. 
450  Ib.  at6f£ 
240  Ib.  at  $1.334 


600yd.  at  370. 
1000  yd.  at  $1.871. 
2000yd.at31.12J. 

296  yd.  at  12|  0. 

180  yd.  at 


1250yd.  at  32  £ 

64yd.  at  $25. 

1644yd.  at  25  £ 

964yd.  at  $2.50 

3000yd.  at  23  £ 


118  FRACTIONS  [§§  322-327 

322.  To  find  the  cost  of  articles  sold  by  the  hundred. 

323.  Example.      What  is  the  cost  of  444  Ib.  of  phosphate   at 
$  2.50  per  hundred  pounds  ? 

SOLUTION.  444  Ib.  equals  4.44  hundred  pounds.  If  1  hundred  Ib.  cost  $2.50, 
4.44  hundred  pounds  will  cost  4.44  times  $2.50,  or  $11.10. 

324.  Hence  the  following  rule : 

Reduce  the  quantity  to  hundreds  and  decimals  of  a  hun- 
dred by  pointing  off  2  places  from  the  right.  Multiply  the 
number  of  hundreds  by  the  price  per  hundred. 

WRITTEN   EXERCISE 
Find  the  cost  of : 

1.  1600  Ib.  salt  at  $1.25  per  G ;  1700  Ib.  at  $1.12J  per  C. 

2.  378  fence  posts  at  $7.50  per  C ;  420  posts  at  $6.66f  per  C. 

3.  905  Ib.  lead  at  $  3.33£  per  C ;  250  Ib.  at  $3.20  per  G. 

4.  1125  Ib.  castings  at  $2.50  per  G ;  2500  Ib.  at  $2.40  per  C. 

5.  1620  handles  at  $7.50  per  C;  250  at  $6.40  per  C. 

6.  5045  Ib.  beef  at  $12.50  per  C ;  1250  Ib.  at  $14.40  per  C. 

7.  24828  Ib.  nails  at  12| ^  per  C ;  37500  Ib.  at  16f  ^  per  C. 

8.  840  Ib.  scrap  iron  at  $1.33  per  C ;  750  Ib.  at  $1.60  per  C. 

9.  3295  Ib.  guano  at  $4.50  per  C ;  .4500  Ib.  at  $5.50  per  G. 

10.   2456  fence  rails  at  $2.50  per  C;  3750  rails  at  $3.33^  per  C. 

325.  To  find  the  cost  of  articles  sold  by  the  thousand. 

326.  Example.     At   $7  per  M,  what  will  be  the  cost  of  1544 
bricks  ? 

SOLUTION.  1544  bricks  equals  1.544  thousand  bricks;  if  1  thousand  bricks 
cost  $7,  1.644  thousand  will  cost  1.544  times  $7,  or  $10.808  =  $10.81. 

327.  Therefore  the  following  rule : 

Reduce  the  quantity  to  thousands  and  decimals  of  a  thou- 
sand by  pointing  off  3  places  from  the  right.  Multiply  the 
number  of  thousands  by  the  cost  per  thousand. 


§§  327-330]  QUANTITY,  PRICE,  AND  COST  119 

WRITTEN   EXERCISE 

Find  the  cost  of : 

1.  1650  ft.  pine  lumber  at  $5  per  M. 

2.  611  ft.  oak  lumber  at  $24  per  M. 

3.  21168  ft.  hemlock  lumber  at  $  7.50  per  M. 

4.  9475  ft.  elm  lumber  at  $13  per  M. 

5.  2120  ft.  ash  lumber  at  $25  per  M. 

6.  2768  ft.  maple  lumber  at  $  14  per  M. 

7.  1100  ft.  chestnut  lumber  at  $  18  per  M. 

8.  4560  ft.  oak  lumber  at  $  22  per  M. 

9.  11265  ft.  spruce  lumber  at  $  12.50  per  M. 
10.    76000  shingles  at  $5.25  per  M. 

328.  To  find  the  cost  of  articles  sold  by  the  ton  of  2000  Ib. 

329.  Example.     What  will  be  the  cost  of  3108  Ib.  of  coal  at  $  6 
per  ton  ? 

SOLUTION.  '3108  Ib.  equals  3.108  half  tons.  Since  1  ton,  or  2000  Ib.,  costs 
$6,  \  of  a  ton,  or  1000  Ib.,  will  cost  \  of  $6,  or  $3.  If  J  of  a  ton  costs  $3,  3.108 
half  tons  will  cost  3.108  times  $3,  or  $9.32. 

330.  Hence  the  following  rule : 

Divide  the  price  of  1  ton  ~by  2  and  the  result  will  be  the 
price  of  1000  Ib. 

From  the  right  of  the  quantity  point  off  3  places  and 
multiply  by  the  price  of  1000  Ib. 

WRITTEN   EXERCISE 
Find  the  cost  of : 

1.  2680  Ib.  at  $2  per  ton.  6.  84,725  Ib.  at  $38  per  ton. 

2.  1345  Ib.  at  $7  per  ton.  7.  15,066  Ib.  at  $120  per  ton. 

3.  4372  Ib.  at  $36  per  ton.  8.  9362- Ib.  at  $  4.50  per  ton 

4.  1135  Ib.  at  $2.50  per  ton.  9.  2040  Ib.  at  $  12.40  per  ton. 
£.  116780  Ib.  at  $34  per  ton.  10.  1115  Ib.  at  $35  per. ton. 


120  FRACTIONS  [§§331-333 

331.  To  find  the  cost  of  products  of  varying  weights  per  bushel 

332.  Examples.     1.   Find  the  cost  of  600  Ib,  clover  seed  at 
$7.50  per  bushel  of  60  Ib. 

SOLUTION.  At  $7.50  per  pound,  the  cost  would  be  600  times  $7.50,  or 
$4500,  but  since  the  price  was  not  $7.50  per  pound,  but  $7.50  per  bushel  of 
60  Ib.,  the  cost  will  be  ^  of  $4500,  or  $75. 

2.   Find  the  cost  of  400  Ib.  seed  at  $1.25  per  bushel  of  14  Ib. 

SOLUTION.  At  $  1.25  per  pound  the  cost  would  be  $500 ;  but  since  the  price 
was  not  $1.25  per  pound,  but  $1.25  per  bushel  of  14  Ib.,  the  cost  would  be  fa 
of  $500,  or  $35.71. 

S.  Find  the  cost  of  6400  Ib.  barley  at  75  j*  per  bushel  of  48  Ib. 

SOLUTION.  At  75^  per  pound  the  cost  would  be  6400  times  75^,  or  $4800; 
but  since  the  price  was  not  75^  per  pound,  but  75^  per  bushel  of  48  Ib.,  the 
cost  will  be  ^  of  $4800.  Dividing  by  48  the  result  is  $100. 

333.  Hence  the  following  rule: 

Multiply  the  number  of  pounds  weight  by  the  price  per 
"bushel  and  divide  the  product  by  the  number  of  pounds  in 
one  bushel. 

WRITTEN   EXERCISE 

Find  the  cost  of : 

1.  2400  Ib.  wheat  at  80^  per  bushel  of  60  Ib. 

&  2560  Ib.  corn  at  65^  per  bushel  of  56  Ib. 

8.  3361  Ib.  barley  at  75^  per  bushel  of  48  Ib. 

£  1768  Ib.  millet  at  9 1  per  bushel  of  45  Itx 

5.  2255  Ib.  oats  at  35^  per  bushel  of  32  Ib. 

6.  2172  Ib.  buckwheat  at  50^  per  bushel  of  48  Ibt 

7.  2761  Ib.  beans  at  $1.25  per  bushel  of  62  Ib. 

8.  2500  Ib.  peas  at  $1.40  per  bushel  of  60  Ib. 

9.  3140  Ib.  Hungarian  grass  seed  at  $2.50  per  bushel  of  45  Ib 
10.  2059  Ib.  red  top  grass  seed  at  90^  per  bushel  of  14  Ib. 


^§334-340]  BILLS   AND    ACCOUNTS  121 

BILLS  AND  ACCOUNTS 

334.  Merchandise  is  any  goods  or  commodities  held  for  the  pur- 
pose of  exchange. 

335.  A  bill  or  invoice  is  a  written  statement  in  detail  of  merchan- 
dise sold,  or  of  services  rendered. 

336.  Outline  of  a  Bill.     The  following  is  an  outline  of  what  a  bill 

should  show : 

1.  The  place  and  date  of  sale. 

2.  The  names  of  the  buyer  and  seller. 

3.  The  terms  of  sale. 

4.  The  distinguishing  marks  and  numbers,  if  any,  placed  on  the 
goods. 

5.  The  quantity,  name,  and  price  of  each  item. 

6.  The  entire  amount  of  the  separate  items. 

7.  All  extra  charges. 

8.  Any  discounts  allowed. 

Formerly  the  term  "invoice"  was  applied  only  to  a  written  statement  of 
merchandise  sold  at  wholesale  or  shipped  to  an  agent  to  be  sold  on  commission. 
Now,  however,  the  terms  "invoice"  and  "bill"  are  used  interchangeably  when 
applied  to  a  detailed  statement  of  goods  bought  or  sold  in  the  course  of  trade. 
The  term  "invoice"  is  never  applied  to  a  list  of  expense  items  or  a  statement 
of  services  rendered.  Thus  we  say,  "an  expense  bill,"  "a  physician's  bill," 
"a  freight  bill,"  "a  bill  of  lading,"  etc. 

337.  A  bill  is  receipted  when  the  words  "Received  payment" 
are  written  at  the  bottom,  and  the  name  of  the  seller  is  signed  either 
by  himself  or  by  some  authorized  person. 

Any  authorized  person  may  receipt  a  bill.  When  any  person  other  than  the 
seller  receipts  a  bill,  he  should  first  sign  the  name  of  the  seller,  and  on  the  next 
line  below  his  own  name  or  initials,  preceded  by  the  word  uby  "  or  "per." 

338.  A  debit  is  that  which  costs  value;  a  credit  is  that  which 
produces  value. 

339.  An  account  is  a  collection,  under  an  appropriate  title,  of 
related  debits  and  credits. 

340.  The  debits  of  an  account  are  written  on  the  left  side ;  the 
credits  on  the  right  side. 


122 


FRACTIONS 


[§§  341-344 


341.  The  ledger  is  the  principal  book  of  accounts.    In  it  are 
entered  in  classified  form  the  debits  and  credits  of  all  business 
transactions. 

342.  An  account  current  is  an  itemized  statement  of  all  the  debits 
and  credits  between  two  persons. 

343.  A  statement  is  a  summary  of  the  debits  and  credits  of  a 
personal  account. 

344.  An  inventory  is  usually  an  itemized  schedule  of  the  prop- 
erty of  an  individual,  firm,  or  corporation  not  shown  on  the  regular 
books  of  account. 

An  inventory  is  usually  made  upon  the  event  of  taking  off  a  balance  sheet, 
of  a  change  in  the  business,  of  the  admission  of  a  partner,  of  the  issue  of  stock, 
or,  in  case  of  embarrassment  or  insolvency,  for  examination  by  creditors,  who 
wish  to  know  the  exact  resources  and  liabilities  of  the  business. 


•Bought  of  LORD  &  TAYLOR 


Terms:. 


/  2.2 


Model  Bill  (Hardware) 

In  the  model  given  above,  the  first  figure  shows  the  list  number  of  the 
article,  the  figures  above  the  horizontal  line  show  the  price,  and  the  figures? 


§344] 


BILLS  AND   ACCOUNTS 


123 


below  the  line  the  number  of  dozens  of  that  special  number  which  were  sold. 
Thus,  the  first  item  in  the  model  means  that  ten  dozen  lanterns  in  all  were 
sold,  of  which  4  dozen  were  #4,  at  $6.76  per  dozen,  and  6  dozen  were  #7, 
at  $6.25  per  dozen. 


New  York,. 


Terms: 


To  SIEGEL,  COOPER  &  CO. 


-X2sr^4/.  SxZtr-r7-t/ 


*t/ '    ft*    &r*    *7J   **r'  <£j~£ 


8fo 


Jf 


4LJ-*     #**     J-J-'        *J*        *0          *J*    J-^J >Z>"^ 


Model  Receipted  BUI  (Dry  Goods) 

In  the  model  given  above  the  number  of  yards  in  the  different  pieces  of 
cloth  is  not  uniform.  Since  the  price  is  so  much  per  yard,  it  is  necessary  to 
list  the  number  of  yards  in  each  piece  as  shown. 

401,  392,  403,  etc.,  in  the  first  item  on  the  bill  are  understood  to  mean  40£, 
89f (i)»  30f,  etc.  4292  equals  the  total  number  of  yards  in  the  10  pieces. 

In  finding  the  total  number  of  yards  in  any  number  of  pieces  the  various 
items  should  not  be  copied  to  another  sheet,  but  should  be  added  horizontally 
as  they  stand.  The  fractions  should  be  added  first,  and  then  the  integers. 


124 


FRACTIONS 


New  York,. 


19^' 


TO  T.  B.  CUNNINGHAM,  ». 


Terms:. 


Model  Receipted  Bill  (Groceries) 

In  the  above  model  the  prices  given  are  free  on  board  cars  New  York  city, 
and  the  shippers  prepaid  the  freight  charges  to  Boston,  Mass.  In  all  such  cases 
the  freight  is  part  of  the  selling  price,  and  is  usually  added  to  the  bill  as  shown 
in  the  model. 

When  goods  are  sold  so  that  all  transportation  charges  fall  upon  the  buyers, 
the  cost  of  cartage  is  also  added  to  the  amount  of  the  bill.  In  certain  lines  of 
business  a  charge  is  also  made  for  the  crates  used  in  packing.  The  above  model 
shows  the  proper  arrangement  for  all  such  additions. 

Had  the  above  bill  of  goods  been  sold  free  on  board  cars  Boston,  Mass.,  and 
had  the  shippers  not  prepaid  the  freight  charges,  these  charges  would  be 
deducted  by  the  consignees  from  the  amount  of  the  bill,  on  the  arrival  of  the 
goods.  The  freight  bill  would  then  be  sent  to  the  shippers  for  credit. 

Any  conditions  as  to  time  of  credit,  manner  of  payment,  or  discount  for 
prepayment  should  always  be  recorded  on  a  bill. 


§344] 


BILLS  AND  ACCOUNTS 


125 


Please  remit  only  by  draft  on  New  York,  Boston, 
or  Philadelphia,  or  by  Post  Office  or  Ezprcx 
Money  Order,  as  the  Clearing  Home  compels 
as  to  pay  eolkeuonchaigeion  local  checks. 

Sold  to  "-^'— — -*^ 


TAYLOR,  WOOD  &  CO, 

DEALERS  IN  PROVISIONS 

69  and  71  Second  Street, 

ina field,  cMass., 


Terms, 
Shipped  via 


Model  Receipted  Bill  (Provisions) 

The  second  item  in  the  above  model  shows  how  gross  weights  and  tares  are 
recorded  in  billing.  The  numbers  to  the  left  of  the  hyphen  are  the  gross  weights, 
and  the  numbers  to  the  right  of  the  hyphen,  the  tares.  Thus,  74-14  is  under- 
stood to  mean  that  the  gross  weight  of  a  tub  is  74  Ib.  and  the  tare  14  Ib.  The 
170  is  the  net  weight  of  the  three  tubs.  In  finding  the  net  weight  the  various 
items  should  not  be  copied  to  another  sheet,  but  should  be  added  horizontally  as 
explained  in  62. 

Bills  on  which  commercial  discounts  are  allowed  should  always  be  arranged 
as  shown  in  the  above  model.  Commercial  discounts  are  fully  explained  on 
pages  198-206. 

In  retail  business,  where  running  accounts  are  kept  with  customers,  a  tran- 
script of  the  charges,  or  of  the  charges  and  credits,  is  made,  giving  items  and 
dates  of  purchases  and  payments.  This  transcript  partakes  of  the  nature  of  both 
statement  and  bill  and  is  called  an  account  current. 


126 


FRACTIONS 


344-345 


Boston,  Mass.,. 


ir.        19 


In  account  with  L  O»  HOLLIS 


-&, 


Model  Statement 

The  above  statement  is  an  abstract  of  W.  L.  Anderson's  account.  On  Jan. 
81  a  statement  was  rendered  showing  a  debit  balance  of  $201.39.  This  amount 
is  taken  as  a  basis  for  the  February  statement,  and  to  it  are  added  the  debit 
items  of  the  ledger  since  Jan.  31,  making  a  total  of  $1984.30.  From  this  total 
is  deducted  the  sum  of  the  credits  hi  the  ledger  since  Jan.  31,  or  $1250,  leaving 
a  debit  balance  of  $734.80.  This  $734.30  will  be  the  first  item  entered  upon  the 
March  statement. 

345.  Statements  are  usually  rendered  the  end  of  each  month. 
By  an  exchange  of  statements  errors  are  less  likely  to  occur,  and 
when  made,  are  more  readily  detected. 


§346] 


BILLS  AND  ACCOUNTS 


127 


346.    Wages  are  usually  calculated  on  the  basis  of  8  or  10  hours 
to  a  day.     In  finding  the  amount  due,  in  order  to  avoid  fractions,  it 


PAY    ROLL.      Weekending 


No.  Names  of  Employees 


MOD.  Tat.  Wed.  ITiur.  Fri.    Sat.   Totals    Rate      Amount 


& 


-74 


/7 


It, 


/z-i 


Model  Pay  Roll 

is  best  to  find  the  total  time  in  hours,  multiply  by  the  rate  per  day 
and  then  divide  by  8  or  10,  carrying  decimals  to  three  places. 

Checks  are  sometimes  used  in  paying  off  employees,  but  more  generally  the 
envelope  system  is  used  and  each  employee  is  paid  in  currency  the  amount  due 
him.  To  pay  off  the  employees  in  this  manner  the  bookkeeper  usually  draws 
from  the  bank  the  exact  amount  of  money  and  just  the  denominations  wanted. 
To  do  this  with  absolute  accuracy  it  is  generally  necessary  for  him  to  classify 
and  record  the  denominations  required  for  the  payment  of  the  amount  due 
each  name  on  the  pay  roll  in  a  manner  similar  to  the  following  : 


BILLS 


FRACTIONAL  CURRENCY 


$20 

$10 

$$ 

$2 

$1 

W? 

25? 

10? 

5? 

1? 

1 

1 
1 
1 
1 
1 
1 

1 
1 
1 
1 

1 

1 
1 
1 

1 

1 

1 

1 
1 

1 

1 

1 
1 

2 

1 

1 
2 

1 
1 

2 
1 
1 
4 
2 

1 

6 

b 

6 

3 

1 

2 

6 

2 

10 

128 


FRACTIONS 


[§  340 


Seconfc  National  iSanfc 

CHELSEA,  MASS. 

Pay  Roll  Memorandum 

E.   W.  FOWLE  & 

require  the  following  : 

CO. 

Pennies    10 

3 
12 
25 
60 
20 

10 
10 
60 
50 
50 

Nickels     2 
Dimes  6 

Quarters  2 
Halves      1 

Dollars    3 
2's  .     6 

5's  5 

10's      6 

20's      1 

121 

80 

The  foregoing  model  shows  what 
should  be  done  to  ascertain  the  de- 
nominations required  to  pay  off  the 
amount  shown  in  the  model  pay  roll, 
page  127.  Many  times  columns  are 
provided  for  this  memoranda  on  the 
right-hand  side  of  the  pay  roll,  im- 
mediately after  the  "Remarks" 
column. 

After  the  amount  of  the  pay  roll 
and  the  denominations  required  have 
been  ascertained,  they  are  written  on 
the  pay  roll  memorandum  as  shown 
in  the  accompanying  model.  This 
memorandum  is  attached  to  a  check, 
payable  to  "Pay  Roll,"  which  is 
taken  to  the  bank  and  cashed. 


WRITTEN  EXERCISE 

Find  the  amount  of  each  of  the  following  bills : 

1 

SAGINAW,  MICH.,  Sept.  1,  1904. 
Messrs.  SAGE  BROS.  &  Co., 
Tonawanda,  N.Y. 

Bought  of  WESTON  &  BROWN- 
TERMS  :   Sight  draft  without  notice  after  30  da. 


26416  ft.  Clear  Pine                   2500  per  M 
146250  ft.  Pine  Plank                1250  per  M 
11670  Cedar  Posts                      1000  per  C 
81275  ft.  Clapboards                  2500  per  M 
71000  Shingles                            410  per  M 
66200  ft.  Pine  Lumber              2500  per  M 
111224  Cedar  R.  R.  Ties             333J  per  C 
31000  ft.  Pine  Boards                166«3  per  M 

§346]  BILLS  AND  ACCOUNTS  129 

f 
WORCESTER,  MASS.,  May  15,  1904. 

Messrs.  R.  E.  BARNES  &  Co., 

Detroit,  Mich. 

Bought  of  OSGOOD,  TOWER  &  Co. 
TERMS  :  Cash 


Case 

No.  of  Yd. 

Price 

Items 

Amount 

#19 

16 

pcs.  Bleached  Cotton 

412  453  411  452  44    441 

471  453  42    423  431  433 

47    44    442 

6J? 

#6 

12 

pcs.  Muslin 

371  323  33    353  341  32 

•352  333  37    38i  38i  36 

10? 

#31 

9 

pcs.  Delaine 

39    402  41i  S93  382  40 

423  441  42 

16§? 

#7 

24 

pcs.  Windsor  Prints 

2F  2?3  253  28    26    222 

24    25    82    312  28    241 

25    272  22    281  241  22 

26    24    312  32    22    212 

G$? 

#21 

21 

pcs.  Merrimac  Prints 

281  32    343  282  26    241 

222  242  262  24    261  33 

282  34    271  30    323  24 

e 

#169 

20 

pcs.    Simpson    Mourning 

6§? 

Prints 

40    38    342  40i  32    40 

412  40    162  40    293  30 

272  192  41i  882  30    43 

42    413 

6|? 

#173 

15 

pcs.  Striped  Denim 

40    42    41    40    38    41 

431  442  451  403  46    38 

401  38s  401 

$$? 

1 

130 


FRACTIONS 

s 


C§346 
BOSTON,  MASS.,  Jan.  3,  1904. 


Messrs.  MARTIN  &  WARREN, 

Milwaukee,  Wis. 
Bought  of  HARRIS  BROS.  &  Co. 

TERMS  :  Interest  after  60  da. 


8 

pcs.  F.  A.  Cambric 

56  521  453  502  62  401  60  511                 22  t 

5 

gro.  Jet  Buttons                                     112$ 

8 

pcs.  P.  D.  Goods 

35  453  552  503  51  62  461  60                  26  ^ 

4 

pcs.  G.  Flannel 

35340402403                                      33^ 

8 

pcs.  V.  Barege 

201  25  242  27  263  22  242  22                 16§  ^ 

6 

pcs.  E.  Lining 

40  622  64  551  452  502                           3^ 

3 

pcs.  B.  Silk 

, 

685866                                               98j* 

CLEVELAND,  OHIO,  Oct.  15,  1904. 
Messrs.  BROWN,  HORTON  &  Co., 

Springfield,  Mass. 
Bought  of  EOBINSON,  CAREY  &  Co. 


TERMS:  Net  cash. 


10 

bbl.  Pork                                               1666§ 

5 

bbl.  Mess  Beef                                        1125 

3 

bbl.  Hams 

- 

275-56  281-60  287-62                       12J  j* 

3 

bbl.  Shoulders 

248-37  252-42  371-40                          8$jZ 

8 

tubs  Lard 

71-14  70-15  69-14                               11  ? 

6 

bkt.  Pork  Loins 

314  301  294  312  302  315                      8Jj* 

§346]  BILLS  AND  ACCOUNTS  131 

WRITTEN  REVIEW 
Find  the  amount  of  each  of  the  following  invoices : 

1.  Thurston  &  Denton,  Buffalo,  N.Y.,  bought  of  Brown  Bros.  &  Co., 
Boston,  Mass.,  Jan.  27,  the  following :  8  pcs.  M.  shirting,  402  411  46* 
512  451  503  43  341,  at  6f ^;  15  pcs.  crash,  613  yd.,  at  6J^;  6  pcs.  C. 
jeans,  502  45  50  551  511  462,  at  5^;  25  doz.  M.  L.  thread  at  59^;  10 
pcs.  R.  print,  41  552  45l  51  46  503  40  562  421  522,  at  6J^;  4  pcs.  N. 
sateen,  551  552  603  502,  at  6|^;  5  gro.  F.  braid  at  $  7.621;  16  doz.  L, 
shirts  at  $7.25;  6  pcs.  T.  R.  prints,  251  353  302  31  211  251,  at  4|^j 
25  cases  E.  batts  at  $  6 ;  20  gro.  S.  P.  buttons  at  49^. 

2.  I.  F.  Hoyt,  Milwaukee,  Wis.,  bought  of  Mann  &  Co.,  of  the 
same  city,  Sept.  4:  10  pcs.  N.  sateen,  552  51  50s  541  56  55  522  53  518 
50,  at  5^;  15  pcs.  T.  A.  flannel,  623  651  61  582  55  631  65s  62  602  63 
563  601  58  622  651,  at  33^;  20  pcs.  E.  gingham,  50  521  51  512  55  603 
621  612  58  552  561  533  51  553  612  61  581  56  542  511,  at  6JJ*;  10  pcs.  B. 
checks,  45  521  412  40  553  502  45  511  42  503,  at  25  £ 

8.  Brown  Bros.  &  Co.,  Maiden,  Mass.,  bought  of  W.  D.  Adams  & 
Co.,  Boston,  Mass.,  June  18 :  20  pcs.  L.  gingham,  582  461  413  381  462 
453  512  55  382  35  373  493  402  51s  44  442  40  371  33s  462,  at  8^;  24  pcs. 
W.  print,  441 463  512  393  412  45  483  51  34s  372  35  362  41s  348  491  372  34 
362  423  48  432  531  381  42,  at  6J^ ;  20  pcs.  E.  lining,  45  541  392  488  462 
382  472  372  453  463  428  443  45s  431  352  542  34s  422  532  44l,  at  4j£ 

4.  Jan.  21,  1904,  J.  H.  Palmer,  Sons  &  Co.,  sold  Morrison,  Price 
&  Long  the  goods  shown  below;  terms,  2%  10  da.,  1%  30  da.,  net 
60  da.  10  doz.  knives  and  forks :  3  doz.  #  5  at  $  8.33 J,  3  doz.  #  7  at 
$6.66},  4  doz.  #9  at  $10;  9  doz.  razors :  3  doz.  #12  at  $9,  3  doz. 
#13  at  $12.50,  3  doz.  #18  at  $16.66f ;  12  doz.  panel  saws:  6  doz. 
#1  at  $15,  4  doz.  #4  at  $21,  2  doz.  #5  at  $25;  4  doz.  nutmeg 
graters:  2  doz.  #1  at  $2.25,  2  doz.  #4  at  $1.75;  6  doz.  pocket 
knives :  2  doz.  #  12  at  $  6,  2  doz.  #  16  at  $  7.50,  2  doz.  #  20  at  $  3.75 ; 
5  doz.  burnished  teapots :  2  doz.  #  1  at  $  6.75,  3  doz.  #  2  at  $7.25 ; 
35  doz.  dippers:  18  doz.  #3  at  $1.75,  10  doz.  #6  at  $1.35,  7  doz. 
#4  at  $1.25;  2}  doz.  wash  boilers  at  $37.75;  f  doz.  #2  kettles  at 
$5.87£;  27  tea  kettles  at  97^;  8£  doz.  padlocks  at  $8.75;  f  doz. 
3-qt.  saucepans  at  $  9.37£ ;  11  doz.  2-qt.  saucepans  at  $  7.85 ;  2£  doz. 
#44  dishpans  at  $2.47;  3f  doz.  #14  cups  at  78f^;  59  faucets  at 
$  1.47 ;  2\  doz.  carpet  stretchers  at  $  2.95 ;  6|  doz.  wrought  wrenches 
at  $12.75;  51  doz.  cast  steel  axes  at  $12.50. 


132  FRACTIONS  [§346 

5.  W.  C.  Blanchard,  Hartford,  Conn.,  bought  of  M.  C.  Woods, 
Utica,  N.Y.,  July  15:   10  pcs.  B.  gingham,  60  61s  501  603  51  618 
611  50  55  51s,  at  8^;  10  doz.  F.  E.  braid,  at  23^;  10  pcs.  B.  checks, 
45  41  551  42  52  402  50  55  518  452,  at  24^;  15  gro.  G.  buttons,  at 
$1.124;  2  Pcs-  T-  A-  flannel,  65  60,  at  30^;  6  pcs.  E.  lining,  40  55l 
452  52  41  501,  at  5^;  5  doz.  L.  L.  gloves,  at   $3.05;  4  pcs.  M. 
sateen,  553  55  50  60s,  at  5J^;  5  gro.  T.  braid,  at  $  7.621;  3  doz.  L. 
shirts,  at  $7.20;   6  pcs.  T.  B.  print,  25  35  303  31  21  251,  at  4f  ^; 
10  cases  E.  batts,  at  $6;  20  gro.  S.  P.  buttons,  at  49^;  4  pcs.  V. 
barege,  20  23  25  25,  at  16f  J*j  7  pcs.  W.  print,  458  51  45  50  462  55 
503,  at  5j£ 

6.  Finu.  the  amount  of  Baker,  Taylor  &  Co.'s  inventory,  Jan.  1, 
with  items  as  follows  :  8  pcs.  F.  A.  cambric,  56  52  45  50  52  54  46 
50,  at  22^;  5  gro.  J.  buttons,  at  $1.12i;  15  pcs.  P.  D.  goods,  55  458 
552  508  51  52  461  50  521  54  482  503  o2  551  50,  at  50^;  4  pcs.  G. 
flannel,  358  40  402  408,  at  25^;  6  pcs.  E.  lining,  40  522  54  551  452 
502,  at  31^;  10  pcs.  V.  barege,  201  25  232  27  268  22  242  22  268  28,  at 
16|  t  ;  10  pcs.  B.  H.  checks,  45  52  55  41  402  51s  511  53  502  46,  at  24  ^  ; 
5  pcs.  W.  prints,  252  318  30  282  27,  at  5jtf  ;  15  pcs.  A.  F.  cashmere, 
621  658  601  63  588  602  562  582  60  622  558  581  608  58  551,  at  19  X;  20  pcs. 
L.  gingham,  45  481  461  448  45s  448  46  44  48  46  42  502  51s  462  471  461 
48  49  451  48,  at 


Bender  the  following  statements  : 

7.  On  Feb.  28,  1904,  the  debits  and  credits  of  Mason  &  Hamlin's 
account  with  Lord  &  Taylor,  Boston,  Mass.,  were  as  follows  :  Debits  : 
Jan.  1,  To  merchandise,  $  900.62  ;  Jan.  27,  To  merchandise,  $  200.56  ; 
Feb.  18,  To  merchandise,   $260.93.     Credits:    Feb.  1,  By  cash, 
$175;  Feb.  15,  By  Cash,  $200. 

8.  On  May  31  the  debits  and  credits  of  Burke,  Fitzsimmons  & 
Hone's  account  with  C.  D.  Gray,  Kochester,  N.  Y.,  were  as  follows  : 
Debits  :  April  15,  To  merchandise,  $  900.46  ;  April  30,  To  merchan- 
dise, $  340.92  ;  May  15,  To  merchandise,  $  135.40.     Credits  :   April 
30,  By  merchandise  returned,  $35.40;  May  15,  By  cash,  $300.90; 
May  20,  By  cash,  $  600. 

9.  The  amount  of  the  model  pay  roll,  page  127,  was  determined 
on  the  basis  of  an  8-hour  daj.    Find  the  amount  on  the  basis  of  a 
10-hour  day. 


§346] 


BILLS   AND   ACCOUNTS 


133 


10.   Find  the  amount  of  the  following  pay  roll  (a)  on  the  basis  of 
an  8-hour  day ;  (b)  on  the  basis  of  a  10-hour  day  : 


NAME 

MON. 

TUBS. 

WED. 

THCB. 

FRI. 

SAT. 

RA.TK 

7 

8 

10 

9 

8 

12 

$2  00 

8 

8 

8 

8 

9 

11 

3  00 

5 

8 

5 

1 

10 

9 

3  50 

Drowne,  William  .... 

9 

7 

9 

8 

8 

8 

3.00 

8 

8 

8 

7 

8 

9 

2  50 

Keyser,  Frederick      .    .    . 

8 

7 

7 

8 

9 

8 

1.75 

Martin,  Charles     .... 

9 

10 

8 

7 

9 

8 

3.00 

Smith,  Martin   

7 

8 

8 

9 

9 

5 

3  00 

Warren,  William  .... 

9 

9 

9 

9 

9 

10 

3.50 

Weeks,  Thomas     .... 

8 

9 

10 

9 

8 

9 

2.50 

11.   Find  the  amount  of  the  following  pay  roll  (a)  on  the  basis  of 
an  8-hour  day ;  (b)  on  the  basis  of  a  10-hour  day  : 


NAME 

MON. 

TlTES. 

WED. 

TlIUR. 

FRI. 

SAT. 

KATE 

Breen,  Mildred      .... 

9 
6 

9 

8 

8 

8 

10 
9 

11 
9 

9 

8 

$3.50 

2.75 

Garret,  Ellen     

7 

7 

8 

9 

9 

9 

3.25 

Cutter,  James    

11 

9 

8 

8 

8 

9 

4  10 

Ernst,  Harry     .     .     .    .    , 

9 

9 

7 

6 

7 

9 

4.00 

Foley,  Maude    

9 

5 

8 

8 

7 

10 

3  25 

Gordon,  Ruth 

8 

7 

8 

8 

8 

8 

2  75 

Healey,  Grace 

7 

8 

6 

8 

6 

5 

1  75 

8 

8 

7 

9 

9 

9 

4  00 

Lang,  James      

8 

8 

9 

10 

8 

11 

4  50 

Penny,  George  A  
Pratt,  Helen  

7 
8 

6 
9 

8 
9 

8 

7 

8 

7 

8 
8 

4.50 
3  50 

Schiller,  Helen  

7 

9 

8 

8 

7 

7 

6  00 

Smith,  Frank 

8 

7 

8 

8 

9 

8 

4  20 

Tuckerman,  Leo    .... 
Walker,  Florence  .... 

8 
8 

5 
9 

9 
9 

8 
9 

7 
8 

9 
9 

5.10 
5.50 

12.  Make  pay  roll  memorandums  for  (a)  and  (b)  in  problem  10 ; 
for  (a)  arid  (b)  in  problem  11.  Assume  that  you  are  to  draw  the 
money  from  City  National  Bank. 


DENOMINATE  NUMBERS 

MEASURES 

347.  Concrete  numbers  in  which  the  unit  has  been  established  by 
law  or  custom  are  called  denominate  numbers.     Numbers  expressed 
in  units  of  the   same   denomination   are   simple  numbers.     Simple 
numbers  having  denominate  units  are  simple  denominate  numbers. 
Numbers  expressed  in  units   of   two   or  more  denominations  are 
compound  numbers,  or  compound  denominate  numbers. 

348.  A  measure  is  a  standard  unit  by  which  quantity  is  estimated. 

Quantity  may  be  length,  breadth,  thickness,  area,  volume,  capacity,  weight, 
value,  time,  number,  or  amount. 

349.  The  principal  measures  are  those  of   Weight,  Extension, 
Time,  Capacity,  Value,  and  Angles. 

350.  A  standard  unit  of  measure  is  a  unit  which  has  been  estab- 
lished by  law  or  custom  as  the  one  by  which  other  units  are  to  be 
adjusted. 

The  Winchester  bushel  has  been  adopted  by  the  United  States  as  the  standard 
unit  for  dry  quantities,  such  as  grain,  seeds,  etc. ;  the  gold  dollar  has  been  estab- 
lished as  the  standard  unit  of  money  value ;  etc. 

351.  A  quantity  is  measured  by  finding  how  many  times  it  con- 
tains any  standard  unit  of  measure. 

The  unit  of  extent  is  the  yard;  of  weight,  the  Troy  pound;  etc.  The  num- 
ber of  yards  in  a  piece  of  cloth  may  be  ascertained  by  applying  the  yard  measure ; 
the  weight  of  a  body,  by  the  use  of  the  pound;  etc. 

MEASURES  OP  WEIGHT 

352.  Weight  is  a  quantity  of  matter  expressed  numerically  with 
reference  to  some  standard  unit. 

353.  The  standard  unit  of  weight  in  the  United  States  is  the 
Troy  pound. 

354.  There  are  three  kinds  of  weight  ID   use:    Troy  Weight, 
Avoirdupois  Weight,  and  Apothecaries'  Weight 

134 


§§  355-357]  MEASURES  135 

Troy  Weight 

355.  Troy  weight  is  used  for  weighing  gold,  silver,  and  jewels. 

TABLE 

24  grains  (gr.)     =  1  pennyweight  (pwt»). 
20  pennyweights  =  1  ounce  (oz.). 
12  ounces  =  1  pound  (lb.). 

Ib.      oz.      pwt.        gr. 
1  =  12  =  240  =  5760. 

The  grains  of  the  Troy,  Avoirdupois,  and  Apothecaries'  weights  are  the 
same. 

The  Troy  pound  is  equal  to  22.7944  cubic  inches  of  pure  water  at  its  greatest 
density,  and  is  identical  with  the  Troy  pound  of  Great  Britain. 

In  weighing  diamonds  and  gems  the  unit  generally  employed  is  the  carat, 
which  is  about  3.2  Troy  grains. 

The  term  carat  is  also  used  to  express  the  fineness  of  gold,  24  carats  being 
pure.  Thus,  a  carat  means  ^  part,  and  gold  18  carats  fine  contains  18  parts 
gold,  or  is  |  pure. 

Avoirdupois  Weight 

356.  Avoirdupois  weight  is  used  for  weighing  all  heavy  articles, 
such  as  groceries,  coal,  provisions,  grain,  and  the  metals,  except  gold 
and  silver. 

357.  The  unit  of  Avoirdupois  weight  is  the  pound,  which  con- 
tains 7000  grains. 

TABLE 

16  ounces  (oz.)  as  1  pound  (lb.). 

100  pounds  as  1  hundredweight  (cwt.). 

20  hundredweight,  or  2000  pounds  =  1  ton  (t.). 

t.    cwt.      lb.  oz. 

1«  20  =  2000  =  82,000. 

"Hundredweight"  and  "pounds"  may  be  read  together  as  pounds,  or 

pounds  may  be  read  as  so  many  hundredths  of  a  hundredweight.  Thus,  17 
hundredweight,  29  pounds,  may  be  read  "1729  pounds,'1  or  "17.29  hundred- 
weight" ;  and  2  tons,  7  hundredweight,  31  pounds,  may  be  read  "2  tons,  7.31 
hundredweight." 


136 


DENOMINATE   NUMBERS 


(.'§§  358-359 


358.  In  Great  Britain  the  ton  equals  2240  pounds.    This  in  the 
United  States  is  called  the  long  or  gross  ton,  and  is  used  in  the  custom- 
houses and  in  wholesale  transactions  in  coal  and  iron. 

LONG  TON  TABLE 

112  pounds  =  1  long  hundredweight  (1.  cwt.). 
2240  pounds  =  1  long  ton  (1.  t.). 

359.  A  great  many  commodities  are  bought  and  sold  by  weight. 
The  weight  of  the  standard  measure  is,  in  some  cases,  uniform 
throughout  the  United  States ;  but  in  others  it  is  regulated  by  State 
statutes.    The  following  table  shows  the  weights  of  the  standards 
frequently  used  in  buying  and  selling  various  commodities. 

TABLE  or  WEIGHTS  OF  PRODUCTS 


COMMODITIES 

STANDARD 
MEASURE 

WEIGHT  IN 
AVOIRDUPOIS 
POUNDS 

EXCEPTIONS 

Barley 

bushel 

48 

Ala.,  Ga.,  Ky.,  Pa.,   47;  Cal.,  50; 

La.,  32. 

Beans 

bushel 

60 

Me.,  62  ;  Mass.,  70. 

Beef 

barrel 

200 

Beets 

bushel 

60 

Butter 

firkin 

100 

Clover  seed 

bushel 

60 

New  Jersey,  64. 

Corn  in  the  ear 

bushel 

70 

Miss.,  72  ;  Ohio,  Ind.,  Ky.,  68. 

Corn  meal 

bushel 

50 

Ala.,   Ark.,   Ga.,   111.,    Miss.,    N.C., 

Tenn.,  48. 

Corn,  shelled 

bushel 

56 

Cal.,  62. 

Fish 

quintal 

100 

Flour 

barrel 

196 

Grain 

cental 

100 

Nails 

keg 

100 

Oats 

bushel 

32 

Ida.,  Ore.,  36  ;  Md.,  26  ;  N.  J.,  Va.,  30 

Onions 

bushel 

60 

Peas 

bushel 

60 

Pork 

barrel 

200 

Md.,  Pa.,  Va.,  56. 

Potatoes 

bushel 

CO 

Rye 

bushel 

56 

Cal.,  54. 

Timothy  seed 

bushel 

45 

Ark.,  60  ;  N.  Dak.,  S.  Dak.,  42. 

Wheat 

bushel 

60 

|§  360-365]  MEASURES  1ST 

Apothecaries'  Weight 

360.  Apothecaries'  weight  is  used  by  physicians  and  druggists  in 
compounding  and  prescribing  medicines. 

TABLE 
20  grains     =  1  scruple  (sc.  or  3). 

8  scruples  =  1  dram  (dr.  or  3). 

8  drains     =  1  ounce  (oz.  or  5  )• 
12  ounces    =  1  pound  (Ib.  or  ft)). 

Ib.      oz.      dr.         sc.  gr. 

1  =  12  =  96  =  288  =  5760. 

The  pound,  ounce,  and  grain  of  this  weight  are  identical  with  those  of  the 
Troy  weight,  but  the  ounce  is  differently  divided. 

Drugs  and  medicines  are  bought  and  sold  at  wholesale  by  Avoirdupois  weight. 

COMPARATIVE  TABLE  OP  WEIGHTS 

1  Troy  pound  =  5760  gr.  ;  1  Troy  ounce  =  480  gr. 

1  Apothecaries'  pound  =  5760  gr. ;  1  Apothecaries'  ounce  =  480  gr. 
1  Avoirdupois  pound     =  7000  gr. ;  1  Avoirdupois    ounce  =  437  J  gr. 
176  Troy  or  Apothecaries'  pounds  =  144  Avoirdupois  pounds. 

MEASURES  OF  EXTENSION 

361.  Extension  is  that  property  of  a  body  by  vvhich  it  occupies  a 
portion  of  a  space.     It  has  one  or  more  of  the  dimensions,  length, 
breadth,  and  thickness,  and  may  therefore  be  a  line,  a  surface,  or 
a  solid. 

362.  Magnitude  is  the  term  applied  to  one  or  more  of  the  dimen- 
sions, length,  breadth,  and  thickness. 

363.  A  line  is  a  magnitude  of  only  one  dimension  —  length. 

364.  A  surface  is  a  magnitude  of  two  dimensions  —  length  and 
breadth. 

365.  A    solid    is    a   magnitude  of   three   dimensions  —  length 
breadth,  and  thickness. 


138  DENOMINATE   NUMBERS  [§§366-370 

366.  The  standard  unit  of  extension  in  the  United  States  is  the 
yard. 

The  standard  yard  prescribed  at  Washington  has  been  fixed  with  the  greatest 
precision.  It  is  determined  by  a  brass  rod  or  pendulum,  which  vibrates  seconds 
in  a  vacuum  at  the  sea  level  at  62°  Fahrenheit,  in  the  latitude  of  London,  Eng. 
This  pendulum  is  divided  into  391,393  equal  parts,  and  360,000  of  these  parts 
constitute  a  yard.  A  copy  of  the  standard,  which  is  identical  with  the  present 
standard  of  Great  Britain,  is  kept  in  each  State  capitol. 

Long  Measure 

367.  Long  measure  is  used  in  measuring  lengths  and  distances. 

TABLE 

12  inches  (in.)          =  1  foot  (ft.). 
8  feet  =  1  yard  (yd.). 

6$  yards  or  16 }  feet  =  1  rod  (rd.). 
820  rods  or  6280  feet  =  1  mile  (ml). 

mL      rd.          yd.          ft.  In. 

1  =  820  =  1760  =  5280  =  63,860. 

The  terms  pole  and  perch  are  sometimes  used  instead  of  rod. 

Formerly  the  mile  was  divided  into  8  furlongs  of  40  rods  each.  The  furlong 
is  now  practically  obsolete. 

The  hand,  used  in  measuring  horses,  is  equal  to  four  inches. 

6280  feet  is  the  legal  mile  in  the  United  States  and  England,  and  hence  is 
sometimes  called  the  statute  mile. 

The  knot,  used  in  navigation,  is  equal  to  1.152$  statute  miles,  or  6086  feet. 
It  is  sometimes  called  a  geographic  mile. 

A  league  is  equal  to  three  knots  or  geographic  miles. 

A  pace  is  equal  to  three  feet,  and  five  paces  approximate  a  rod. 

The  fathom,  used  in  measuring  depths  at  sea,  is  equal  to  six  feet 

Square  Measure 

368.  Square  measure  is  iised  in  computing  surfaces  such  as  land, 
floors,  boards,  walls,  and  roofs. 

369.  A  square  is  a  flat  surface  bounded  by  four  equal  sides  and 
having  four  square  corners. 

370.  The  unit  of  square  measure  is  a  square,  each  side  of  which 
is  bounded  by  a  unit  of  length ;  as,  a  square  inch,  a  square  yard. 


§§  370-373]  MEASURES  139 


A  square  inch  is  a  square,  each  side  of  which  is 
one  inch  ;  a  square  foot  is  a  square,  each  side  of 
which  is  one  foot ;  a  square  yard  is  a  square,  each 
side  of  which  is  one  yard  ;  a  square  rod  is  a  square, 
each  side  of  which  is  one  rod. 


One  Square 
Inch 


llnch 


371.  The  area  of  a  figure  is  the  number  of  square  units  contained 

in  its  surface. 

TABLE 

144  square  inches  (sq.  in.)  =1  square  foot  (sq.  ft.). 

9  square  feet  =  1  square  yard  (sq.  yd.). 

80£  square  yards,  or  272 £  square  feet  =  1  square  rod  (sq.  rd.). 
160  square  rods,  or  43,560  square  feet  =  1  acre  (A.). 
640  acres  =  1  square  mile  (sq.  mi.). 

8q.  ml.     A.          sq.  rd.  sq.  yd.  sq.ft.  sq.  in. 

1  =  640  =  102,400  =  3,097,600  =  27,878,400  =  4,014,489,600. 

All  the  units  of  square  measure  except  the  acre  are  derived  from  the  corre- 
sponding units  of  long  measure.  Thus,  144  (12  x  12)  square  inches  =  1  square 
foot ;  9  (3  x  3)  square  feet  =  1  square  yard ;  30J  (5£  x  5£)  square  yards  = 
1  square  rod ;  102,400  (320  x  320)  square  rods,  or  640  acres,  which  are  equal  to 
1  square  mile. 

Surveyors9  Long  Measure 

372.  Surveyors'  long  measure  is  used  by  surveyors  in  measuring 
land,  laying  out  roads,  establishing  boundaries,  etc. 

373.  The  unit  of  surveyors'  long  measure  is  the  Gunteifs  chain, 
which  is  4  rods,  or  66  feet,  in  length. 

The  chain  has  100  links,  which  may  be  written  as  hundredths  of  a  chain. 
Thus,  4  chains,  27  links  =  4.27  chains. 

TABLE 
7.92  inches  =  1  link  (L). 

25  links    =  1  rod. 

4  tods,  or  100  links    =  1  chain  (ch.). 
80  chains  =  1  mile  (ml.). 

mi.      ch.       rd.  1.  in. 

1  =  80  =  320  =  8000  =  63,360. 


140 


DENOMINATE   NUMBERS 


L§§  S74-37& 


Surveyors'  Square  Measure 

374.  Surveyors'  square  measure  is  used  by  surveyors  in  measuring 
land  by  acres  and  sections.    It  is  sometimes  called  land  measure. 

375.  The  unit  of  land  measure  is  the  acre. 

TABLE 
026  square  links  (sq.  11.)  =  1  square  rod  (sq.  rd.). 

16  square  rods  =  1  square  chain  (sq.  ch.). 

10  square  chains,  or  160  square  rods  =  1  acre  (A.). 
640  acres  =  1  square  mile  (sq.  mi). 

In  some  parts  of  the  country  36  square  miles,  or  6  miles  square,  is  a  township. 
A  square  mile,  or  640  acres,  is  also  called  a  section  in  surveying  public  lands. 

sq.  mi  A.  sq.  ch.  sq.  rd.  sq.  1. 


1          = 


640 


6400 


10,2400 


64,000,000. 


376.  United  States  public  lands  are  surveyed  by  selecting  a  north 
and  south  line  as  a  principal  meridian,  and  an  east  and  west  line 
intersecting  this  as  a  base  line.    From  these,  other  lines  are  run  at 
right  angles,  six  miles  apart,  thus  dividing  the  territory  into  townships 
six  miles  square. 

377.  The  rows  of  townships  running  north  and  south  are  called 
ranges.    The  townships  in  each  range  are  numbered  north  and  south 
of  the  base  line,  and  the  ranges  axe  numbered  east  and  west  from 
the  principal  meridian. 

378.  Each  township  is  divided  into  36  sqtiares  of  1  square  mile 
each.     These   squares 

are  called  sections,  and 
are  divided  into  halves 
and  quarters ;  each 
quarter  section  is  in 
turn  divided  into  halves 
and  quarters. 

379.  The  numbering  of  the  sections  in  every  township  is  as  in 
the  accompanying  diagram.     The  corners  of  all  sections  are  perma- 
nently marked  by  monuments  of  stone  or  wood,  and  a  description  of 
the  monument  and  its  location  is  made  in  the  field  notes  of  the 
surveyor. 


ATOWNSHIP 


A  SECTION 


W- 


N.  i  Section 
(320  A.) 

8.W.J 
(160  A.) 

W.i 

of 
S.E.J 

(80  A.) 

*£* 

S.E.J 

S.E.i 
8.1* 

§§  380-385]  MEASURES  141 

Cubic  Measure 

380.  Cubic  measure  is  used  in  determining  the  contents  or  vol- 
ume of  solids. 

381.  A  cube  is  a  regular  solid  bounded  by  six  equal  square  sides 
or  faces.     Its  length,  breadth,  and  thickness  are,  therefore,  equal. 

382.  The  unit  of  cubic  measure  is  a  cube,  each  side 
of  which  is  bounded  by  a  unit  of  length ;  as,  a  cubic 
inch,  a  cubic  yard. 


A  cubic  inch  is  a  cube,  each  side  of  which  is  one  inch  ;  a 
cubic  foot  is  a  cube,  each  side  of  which  is  one  foot;  a  cubic 
yard  is  a  cube,  each  side  of  which  is  one  yard.  Cubic  Foot. 

383.  The  contents  or  volume  of  a  cubical  body  is  the  number  of 
cubic  units  it  contains. 

TABLE 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.). 
27  cubic  feet  =  1  cubic  yard  (cu.yd.). 

cu.yd.  cu.  ft.       cu.in. 
1   =  27  =  46,656. 

The  units  of  cubic  measure  are  derived  from  the  corresponding  units  of  long 
measure.  Thus  1728  (12  x  12  x  12)  cubic  inches  =  1  cubic  foot ;  27  (3  x  3  x  3) 
cubic  feet  =  1  cubic  yard. 

A  cubic  foot  of  water  contains  nearly  7£  gallons  and  weighs  1000  ounces,  or 
62|  Avoirdupois  pounds.  Hence,  a  gallon  of  water  weighs  about  8|  pounds. 

WOOD  TABLE 

16  cubic  feet  =  1  cord  foot  (cd.  ft.). 

8  cord  feet,  or  128  cubic  feet  =  1  cord  (cd.). 

A  cord  of  wood  is  a  pile  8  feet  long,  4  feet  high,  and  4  feet  thick. 
A  perch  of  masonry  is  16£  feet  long,  1£  feet  wide,  and  1  foot  high,  and  con- 
tains 24|  cubic  feet. 

A  cubic  yard  of  earth  is  called  a  load. 

MEASURES  OF  CAPACITY 

384.  Capacity  signifies  room  for  things. 

385.  There  are  two  measures  of  capacity  in  general  use  —  dry 
measure  and  liquid  measure. 


142  DENOMINATE   NUMBERS  [§§  386-389 


Dry  Measure 

386.  Dry  measure  is  used  in  measuring  grain,  fruit,  vegetables, 
coal,  etc. 

387.  The  unit  of  dry  measure  is  the  Winchester  bushel,  which  is 
8  inches  deep,  18^  inches  in  diameter,  and  contains  2150.42  cubic 
inches. 

This  is  the  standard  unit  in  uniform  use  for  measuring  shelled  grains.  The 
heaped  bushel  of  2747.71  cubic  inches  is  used  for  measuring  apples,  roots,  corn 
in  the  ear,  etc. 

The  British  imperial  bushel  contains  2218.19  cubic  inches. 

TABLE 

2  pints  (pt.)  =  1  quart  (qt.). 
8  quarts  =  1  peck  (pk.). 
4  pecks  =  1  bushel  (bu.). 

bu.  pk.     qt.      pt. 
1=4  =  32  =  64. 

The  dry  gallon,  or  half  peck,  contains  268.8  cubic  inches. 


Liquid  Measure 

Liquid  measure  is  used  in  measuring  liquids  and  in  estimat- 
ing the  capacity  of  cisterns,  reservoirs,  etc. 

389.    The  unit  of  liquid  measure  is  the  gallon,  which  contains 

231  cubic  inches. 

TABLE 

4  gills  (gi.)  =  l  pint  (pt.). 
2  pints         =  1  quart  (qt.). 
4  quarts       =  1  gallon  (gal.). 
81$  gallons     =  1  barrel  (bbl.). 

gal.    qt.    pt.     gi. 
1  =  4  =  8  =  32. 

Casks,  called  hogsheads,  pipes,  butts,  etc.,  are  not  fixed  measures,  their 
capacity  varying  for  commercial  purposes. 

In  the  sale  of  oils  and  liquors,  and  in  certain  other  cases,  the  barrel  is  also 
an  indefinite  quantity. 


§§  390-394] 


MEASURES 


143 


Apothecaries'  Fluid  Measure 

390.   Apothecaries'  fluid  measure  is  used  in  measuring  the  liquids 
used  in  compounding  medical  prescriptions. 

TABLE 

60  minims  (Tt\,)    =  1  fluid  drachm  (f.  3). 
8  fluid  drachms  =  1  fluid  ounce  (f.  5). 
16  fluid  ounces    =  1  pint  (O). 

COMPARATIVE  TABLE  OF  MEASURES 


MEASURE 

Cir.  IN.  IN 
ONE  GAL. 

Cu.  IN.  IN 
ONE  QT. 

Cu.  IN.  IN 
ONE  Pr. 

Cu.  IN.  IN 
ONE  Gi. 

Dry 
Liquid 

G  pk.)  268f 
231 

67* 
57f 

33§ 
98} 

8* 

7& 

MEASURES  OF  TIME 

391.  Time  is  a  measure  of  duration.     Its  computations  being 
based  upon  planetary  movements  are  the  same  in  all  lands  and 
among  all  peoples, 

392.  The  unit  of  time  is  the  solar  day;  it  includes  one  revolution 
of  the  earth  on  its  axis,  and  is  divided  into  24  hours,  counting  from 
midnight  to  midnight  again. 

393.  A  solar  year  is  the  exact  time  required  by  the  earth  to  make 
one  complete  rotation  around  the  sun,  —  365  days,  5  hours,  48  min- 
utes, 49.7  seconds,  or  about  365^  days. 

394.  The  solar  year  is  divided  in  the  calendar  into  365  days 
called  a  common  year,  except  every  fourth  year,  when  one  day  is 
added  to  the  month  of  February  and  the  year  is  called  a  leap  year. 
Since  the  fraction  that  is  disregarded  when  365  days  is  counted  as  a 
year  is  less  than  one  fourth  of  a  day,  the  addition  of  a  day  every 
fourth  year  is  not  exactly  accurate.     The  slight  error  still  existing 
is  corrected  by  excluding  from  the  leap  years  the  centennial  years 


144  DENOMINATE   NUMBERS  [§394 

which  are  not  divisible  by  400.     Hence,  to  find  whether  any  year  is. 
a  leap  year  or  not, 

Divide  the  number  of  centennial  years  by  400  and  all  other  years  by 
4 ;  if  there  is  no  remainder,  the  year  is  a  leap  year. 

TABLE 

60  seconds  (sec.)  =  1  minute  (min.). 
60  minutes  =  1  hour  (hr.). 

24  hours  =  1  day  (da.). 

7  days  =  1  week  (wk.). 

365  days  =  1  common  year  (yr.). 

866  days  =  1  leap  year. 

100  years  =  1  century  (C.). 

yr.     mo.          da.          hr.  min.  sec. 

I  _  12  _  /  365  =  8760  =  525,600  =  31,536,000. 
1 366  =  8784  =  527,040  =  31,622,400. 

COMMERCIAL  TABLE 
30  days       =  1  month  (mo.). 
12  months  =  1  year  (yr.). 

The  12  months  into  which  we  divide  the  year  are  called  calendar  months. 
They  are  of  variable  length,  seven  of  them  containing  31  days,  four  30  days, 
and  February  28  days,  except  in  leap  years,  when  it  has  29  days. 

The  calendar  months,  with  the  number  of  days  they  contain,  are  shown 
below : 

1.  January  (Jan.)  31  da.  7.  July  31  da. 

2.  February  (Feb.)  28-9  da.        8.   August  (Aug.)  31  da. 

3.  March  (Mar.)  31  da.  9.   September  (Sept.)  30  da. 

4.  April  (Apr.)  30  da.          10.   October  (Oct.)  31  da. 

5.  May  31  da.          11.   November  (Nov.)  30  da. 

6.  June  30  da.          12.   December  (Dec.)  31  da. 

Standard  Time.  In  1883  the  principal  railroads  of  the  United  States  and 
Canada  adopted  what  is  known  as  the  "  Standard  Time  System."  This  system 
divides  the  United  States  and  Canada  into  four  sections  or  time-belts,  each 
covering  15°  of  longitude,  7£°  of  which  are  east  and  7|°  west  of  the  governing 
or  standard  meridian,  and  the  time  throughout  each  belt  is  the  same  as  the 
astronomical  or  local  time  of  the  governing  meridian  of  that  belt.  The  govern- 
ing meridians  are  the  75th,  the  90th,  the  105th  and  the  120th  west  of  Greenwich, 
and  as  these  meridians  are  just  15°  apart,  there  is  a  difference  in  time  of  exactly 
one  hour  between  any  one  of  them  and  the  one  next  on  the  east,  or  the  one 
next  on  the  west ;  the  standard  meridian  next  on  the  east  being  one  hour  faster, 
and  the  one  next  on  the  west  one  hour  slower.  The  time  of  the  75th  meridian  is 
called  Eastern  Time.  The  time  of  the  90th  meridian  is  known  as  Central  Time. 


§§  394-399]  MEASURES  145 

The  time  of  the  105th  meridian  is  known  as  Mountain  Time.  Time  in  the  fourth 
belt,  which  is  governed  by  the  120th  meridian,  and  extends  to  the  Pacific  coast, 
is  called  Western  or  Pacific  Time.  The  changes  from  one  time  standard  to 
another  are  made  at  the  termini  of  roads,  or  at  well-known  points  of  departure, 
and  where  they  are  attended  with  the  least  inconvenience  and  danger.  Thia 
system  has  been  adopted  by  most  of  the  principal  cities  for  local  use. 

MEASURES  OF  VALUE 

395.  Value  is  the  worth  of  one  thing  as  compared  with  another. 
The  general  measure  of  value  is  money. 

United  States  Money 

396.  United  States  money  has  been  fully  treated  on  pages  58  to 
64  inclusive. 

Canadian  Money 

397.  Canadian  money  is  the  legal  currency  of  the  Dominion  of 
Canada;    it  consists  of  gold,  silver,  and  bronze  coins,  and  paper 


10  mills  (m.)  =  1  cent  (^). 
100  cents          =  1  dollar  ($). 

The  mill  is  not  coined.  The  gold  coins  of  Canada  are  the  British  sovereign 
and  half-sovereign  ;  the  silver  coins  are  the  5,  10,  25,  and  50-cent  pieces  ;  the 
only  bronze  coin  is  the  cent.  The  Canadian  silver  coins  are  f  £  pure  metal  and 
&  copper. 

English  Money 

398.  English  or  sterling  money  is  the  legal  currency  of  Great 
Britain  ;  it  consists  of  gold,  silver,  copper,  and  bills. 

399.  The  unit  of  English  money  is  the  pound  sterling,  the  value 
of  which  in  United  States  money  is  $4.8665. 

TABLE 

4  farthings  (far.)  =     1  penny  (d.). 
12  pence  =     1  shilling  (s.). 

20  shillings 


1  pound 

£         8.  d.         far. 

1  =  20  =  240  =  960. 

«f.,  s.,  £,  are  the  initial  letters  of  the  Latin  words  denarius,  solidarius,  libra, 
signifying  respectively,  penny,  shilling,  and  pound. 


146  DENOMINATE   NUMBERS  [§§400-403 

400  The  value  in  United  States  money  of  the  different  denomi- 
nations of  English  money  is  shown  in  the  following 

COMPARATIVE  TABLE 

1  farthing  =  $  cent.  1  shilling  =  24  J  cents. 

1  penny     =  2s2g  cents.  1  pound   =  $4.8665. 

The  farthing  is  but  little  used  except  as  a  fractional  part  of  the  penny. 

The  British  gold  coins  are  }}  pure  gold  and  ^  alloy  ;  the  silver  coins,  f  £ 
pure  silver  and  ?%  copper  ;  the  penny  and  half-penny  pieces  are  pure  copper. 

The  gold  coins  are  the  sovereign  and  the  half-sovereign  ;  the  silver  coins  are 
the  crown  (equal  to  5  shillings),  the  half-crown,  the  florin  (equal  to  2  shillings), 
the  shilling,  the  six-penny  piece,  the  four-penny  piece,  and  the  three-penny 
piece ;  the  copper  coins  are  the  penny,  the  half-penny,  and  the  farthing.  The 
guinea  (equal  to  21  shillings),  and  the  half-guinea  are  no  longer  coined. 

French  Money 

401.  French  money  is  the  legal  currency  of  France ;  it  is  a  deci- 
mal currency,  and  consists  of  gold,  silver,  and  bronze  coins   and 
national  bank  notes. 

402.  The  unit  of  French  money  is  the  franc,  which  is  equal  to 
19.3  cents  in  United  States  money. 

The  franc  is  also  used  in  Belgium  and  Switzerland,  and  under  different  names 

in  several  other  countries. 

TABLE 

10  millimes  (m.)  =  1  centime  (c.). 
10  centimes  =- 1  decime  (dc.). 
10  declines  =  1  franc  (fr.) 

fr.     do.        c.  m. 

1  =  10  =  100  =  1000. 

403.  The  value  in  United  States  money  of  the  different  denomi- 
nations of  French  money  is  shown  in  the  following 

COMPARATIVE  TABLE 
1  centime  =  $.00193.  1  decime  =  $.0193.  1  franc  =  $.193. 

The  millime  is  not  a  coin.  The  gold  coins  of  France  are  the  5,  10,  50,  and 
100  franc  pieces,  which  are  -fa  pure  gold  and  j^  alloy  ;  the  silver  coins  are  the 
1,  2,  and  6  franc  pieces  ;  also  the  25  and  60  centime  pieces ;  they  are  ft  pure 
and  -jij  alloy.  The  bronze  coins  are  the  1,  2,  5,  and  10  centime  pieces. 

French  money  is  read  as  francs  and  centimes  in  the  same  manner  as  United 
States  money  is  read  dollars  and  cents. 


§§  404^12]  MEASURES  147 

German  Money 

404.  German  money  is  the  legal  currency  of  the  German  Empire; 
it  consists  of  gold,  silver,  and  nickel  coins,  and  paper  money. 

405.  The  unit  of  German  money  is  the  mark,  which  is  equal  to 
23.85  cents  in  United  States  money. 

TABLE 
100  pfennige  (Pf.)  =  1  mark  (Rm.). 

The  gold  coins  of  the  German  Empire  are  the  5,  10,  and  20  mark  pieces ; 
the  silver  coins  are  the  1  and  2  mark  pieces  and  the  20  and  50  pfennig  pieces ; 
the  nickel  coins  are  the  5  and  10  pfennig  pieces  ;  the  copper  coins  are  the  1  and 
2  pfennig  pieces.  The  gold  and  silver  coins  are  T9ff  pure  and  T^  alloy. 


ANGULAR  MEASURE 

406.  Angular  measure  is  used  in  surveying,  civil  engineering, 
astronomical   calculations,    and   navigation,  for   measuring   angles, 
determining  directions  and  location  of  places,  latitude,  longitude, 
difference  in  time,  etc. 

407.  The  unit  of  angular  measure  is  the  degree,  which,  in  any 
circle,  is  measured  by  -^-^  of  the  circumference. 


408.  A  circle  is  a  plane  figure  bounded  by  a  curved  line,  every 
point  of  which  is  equally  distant  from  a  point  within,  called  the 
center. 

409.  The    circumference  of   a  circle  is   the 
curved  line  bounding  it. 

410.  The  diameter  of  a  circle  is  a  straight 
line  passing  through  the  center  and  having  its 
end  in  the  circumference. 

411.  The  radius  of  a  circle  is  a  straight  line  passing  from  the 
center  to  any  point  in  the  circumference. 

412.  Any  part  of  the  circumference  of  a  circle  is  called  an  arc. 


148  DENOMINATE   NUMBERS  [§413 

413.  Every  circle,  great  or  small,  is  divisible  into  4  parts  called 
quarters,  which  are  divisible  into  90  equal  parts  called  degrees.  Every 
circle  therefore  may  be  divided  into  360  degrees. 

TABLE 

60  seconds  (")  =  1  minute  ('). 
60  minutes        =  1  degree  (°). 
860  degrees         =  1  circle  (Cir.). 
.       Cir.      o  »  •» 

1  =  360  =  21,600  =  1,296,000. 

Minutes  of  the  earth's  circumference  are  called  nautical  or  geographic  miles  ; 
hence,  a  degree  of  the  earth's  surface  at  the  equator  contains  60  geographic 
miles,  or  69J  statute  miles. 

MISCELLANEOUS  MEASURES 

ENUMERATION  TABLE 
12  units  =  1  dozen  (doz.). 
12  dozen  =  1  gross  (gro.). 
12  gross  =  1  great  gross  (gt.  gro.). 

gt.  gro.    gro.         doz.         units. 
1  =  12  =  144  =  1728. 

Two  units  are  often  called  a  pair •,  and  20  units  a  score. 

STATIONERS'  TABLE 
24  sheets  (sht.)  =  1  quire  (qr.). 
20  quires  =  1  ream  (rm.). 

2  reams  =  1  bundle  (bdl.). 

6  bundles         =  1  bale  (bl.). 

bL    bdl.     rm.       qr.         sht. 
1  =  6  =  10  =  200  =  4800. 


DRILL  EXERCISE 

1.  Write  two    like    concrete    numbers;     two   unlike   concrete 
numbers. 

2.  Write  two  simple  denominate  numbers;   two  compound  de- 
nominate numbers. 

8.   Write  two  like  denominate  numbers ;  two  unlike  denominate 
numbers. 


§  413]  MEASURES  149 

4.  Which  is  the  heavier,  a  Troy  pound  or  an  Apothecaries' 
pound?  a  Troy  pound  or  an  Avoirdupois  pound?     Illustrate. 

5.  Which  is  the  heavier,  a  Troy  ounce  or  an  Avoirdupois  ounce  ? 
Illustrate. 

6.  Write  two  numbers  expressing  the  weight  of  a  dry  medical 
prescription ;  of  gold ;  of  hay ;  of  vinegar ;  of  wheat ;  of  feathers. 

7.  Write  a  number  expressing  volume ;  surface ;  distance. 

8.  Write  a  number  expressing  Canadian  money  value ;  United 
States  money  value ;  French  money  value ;  German  money  value. 

9.  Write  a  number  expressing  the  weight  of  a  liquid  medical 
prescription. 

10.  Write  three  articles  that  are  sold  by  the  enumeration  table. 

11.  Write  a  number  expressing  a  quantity  of  apples  in  storage 
in  a  certain  warehouse ;  potatoes ;  onions ;  beef ;  fish. 

12.  Write  a  number  expressing  surveyors'  square  measure ;  an- 
gular measure ;  surveyors'  long  measure ;  cubic  measure. 

13.  Write  five  numbers  expressing  area;    three  expressing  ca 
pacity ;  two  expressing  time. 

14.  Write  the  next  leap  year ;  the  next  centennial  year. 

15.  Write  the  standard  unit  of  weight  in  the  United  States  in 
grains. 

16.  Write  the  standard  unit  of  English  money ;  of  United  States 
money ;  of  French  money ;  of  German  money. 

17.  Write  the  standard  unit  of  dry  measure  in  cubic  inches. 

18.  Write  a  dry  gallon  in  cubic  inches ;  a  liquid  gallon ;  a  heaped 
bushel. 

19.  Write  a  degree  of  the  earth's  surface  at  the  equator  in  statute 
miles ;  in  geographic  miles. 

20.  Write  a  perch  of  masonry  in  cubic  inches;  a  cord  of  wood  in 
cubic  feet ;  a  township  in  square  miles. 

21.  Express  in  United  States  money  the  difference  between  a 
franc  and  a  quarter  of  a  dollar ;  the  difference  between  a  pound  ster- 
ling and  $  10. 

22.  A  man  has  4  English  sovereigns,  3  half-dollars,  and  5  francs. 
Express  the  total  sum  in  United  States  money. 


150  DENOMINATE  NUMBERS  [§§413-416 

23.   Express  5  reams  of  paper  in  sheets ;  5  quires. 
24*   Express  a  statute  mile  in  feet ;  in  inches. 
25.   Express  in  Canadian  money  the  cost  of  11  gross  of  lead  pen 
cils  at  30  ^  a  dozen. 

DENOMINATE  QUANTITIES 
REDUCTION  OP  DENOMINATE  INTEGERS 

414.  In  reduction  the  unit  or  denomination  of  a  number  changes, 
but  not  the  value.     When  the  change  is  from  a  higher  to  a  lower 
denomination  the  process  is  called  reduction  descending)  and  when 
from  a  lower  to  a  higher,  reduction  ascending. 

DRILL  EXERCISE 

1.  Find  the  cost  of  90  ft.  of  cable  at  $  1.20  per  yard. 

2.  What  will  5  sq.  ft.  of  gold  leaf  cost  at  2^  per  square  inch  ? 
8.   How  many  inches  in  3  yd.  2  in.  ? 

4.  How  many  five-cent  pieces  should  be  given  for  an  eagle  ? 

5.  How  many  units  in  2  gro.  ? 

6.  Find  the  cost  of  2  bu.  of  apples  at  100  a  peck. 
T  In  -^  of  an  acre  how  many  square  feet  ? 

8.  in  -fa  sq.  mi.  how  many  square  rods  ?  in  -^  sq.  mi? 

9.  In  288  Avoirdupois  pounds  how  many  Troy  pounds  ?    How 
many  Apothecaries'  pounds  ? 

10.   Express  in  English  money  $486.65;    in    French    money 
$19.30;  in  German  money  $23.85. 

415.  Reduction  from  a  higher  denomination  to  a  lower. 

416.  Example.    Reduce  £45  5s.  Sd.  to  pence. 

£  45  5s    Sd  SOLUTION.    Since  £1  Is  equal  to  20s.,  £45  are  equal 

to  45  times  20s.,  or  900s.    900s.  with  the  5s.  added  is 
^—  equal  to  905s. 

905s.  Since  in  Is.  there  are  12&,  in  905s.  there  are  905 

12  times  12d.,  or  10,860d.     10,860A  with  the  Sd.  added  is 

10868c?.  equal  to  10,868d.,  or  the  required  result. 

When  possible,  add  mentally  the  number  of  lower 
denomination  to  the  product  as  shown  in  the  illustration  in  the  margin. 


§§417-41'.)]  DENOMINATE    QUANTITIES  151 

417.  From  the  foregoing  illustration  the  following  rule  may  be 
derived  : 

Multiply  tTie  units  of  the  highest  denomination  given  by 
the  number  of  the  next  lower  denomination  required  to 
make  one  of  this  higher,  and  to  the  product  add  the  given 
units,  if  any,  of  the  lower  denomination. 

Proceed  in  this  manner  with  each  successive  result  until 
the  required  denomination  is  reached. 

WRITTEN  EXERCISE 
Reduce  to  the  lowest  denomination  named : 

1.  3  mi.  17  rd.  3  yd.  1  in.  11.   9  T.  5  cwt.  4  Ib.  1  oz. 

2.  £51  10s.  3d  12.   5  Ib.  8  oz.  13  pwt. 

3.  11  bu.  5  pt.  13.    19  rd.  5  in. 

4.  5  bu.  1  pk.  7  qt.  14.   14  sq.  rd.  5  sq.  yd.  3  sq.  ft. 

5.  43°  T  23".  15.   4  A.  31  sq.  rd.  5  sq.  yd.  3  sq.  ft. 

6.  17  gal.  2  qt.  1  gi.  16.   5  wk.  367  hr.  5  inin.  31  sec. 

7.  5  1. 1.  50  Ib.  2  oz.  17.   7  en.  yd.  5  cu.  ft. 

8.  5  cu.  yd.  3  cu.  ft.  11  cu.  in.         18.    7  Ib.  3  pwt.  4  gr. 

9.  175  sq.  rd.  15  sq.  in.  19.    1  bbl.  2  gal.  1  pt. 

10.   1  mi.  15  ch.  43  li.  20.   2  mi.  15  rd.  11  ft.  10  in. 

418.  Reduction  from  a  lower  denomination  to  a  higher. 

419.  Example.     Keduce  473  pt.  to  bushels. 

2 1 473  pt.  SOLUTION.     Since  2  pt.  equal  1  qt.,  8  qt.  1  pk., 

8  236  qt    +1  pt  anc*  4  Pk*  1  bu-'  tne  successive  divisors  for  re- 

4-  ~ 9Q    lr  _t-  A.    f  ducing  given  pints  to  bushels  are  2,  8,  and  4, 

— -  £  respectively. 

1  Pk<  Divide  473  pt.  by  2  and  the  result  is  236  qt. 

473  pt.  =  7  bu.  1  pk.       wjth  a  remainder  1  pt.  ;  divide  236  qt.  by  8  and 

4  qt.  1  pt.         the  result  is  29  pk.  with  a  remainder  4  qt. ;  divide 

29  by  4  and  the  result  is  7  bu.  with  a  remainder  1  pk. 

Write  the  last  quotient  and  the  several  remainders  in  order  and  the  required 
result  is  7  bu.  1  pk.  4  qt.  1  pt. 


152  DENOMINATE   NUMBERS  £§§420-42,] 

420-   Hence  the  following  rule  : 

Divide  the  given  number  by  the  number  of  the  same 
denomination  required  to  make  one  of  the  next  higher 
denomination,  and  consider  the  quotient  as  units  of  the 
higher  denomination,  and  the  remainder  as  units  of  the 
lower  denomination. 

Proceed  in  like  manner  with  each  successive  quotient 
until  the  required  denomination  is  reached. 

The  last  result  and  the  several  remainders  written  in  order 
will  be  the  answer  required. 

WRITTEN  EXERCISE 

Change  to  units  of  higher  denominations  : 
1.  72,920  min.  5.  214,712  in.  9.  9537  sec. 

&  24,840  gi.  6.  60,720  oz.  Avoir.  10.  10,632  sq.  rd. 

3.  7210  pt.  dry  meas.     7  52,460  gr.  Troy.  11.  8792  cu.  in. 

4.  40,720  sq.  yd.  8.  24,620  da.  12.  34,832  Ib.  Avoir. 

REDUCTION  OF  DENOMINATE  FRACTIONS 

421.  When  the  integral  unit  of  a  fraction  is  a  denominate  num- 
ber, the  fraction  is  called  a  denominate  fraction. 

422.  Reduction  of  denominate  fractions  from  a  higher  denomination 
to  a  lower. 


423.   Examples.    1.  Reduce  j^  Troy  pounds  to  the  fraction  of 

a  pennyweight. 

SOLUTION.  Denominate  fractions  may 
be  reduced  to  lower  denominations  by 
multiplication  in  practically  the  same 
'  X  If  x  ^-  =  —  p^.  manner  as  denominate  integers. 

1        1       31  Since  12  oz.  equal  a  pound,  and  20 

pwt.  equal  1  oz.,  the  successive  multipli- 
ers  for  reducing  pounds  to  pennyweights 
are  12  and  20  respectively. 

Multiplying  r^30  by  12  and  20,  by  can 
cellation  the  result  is  found  to  be  ff  pennyweight 


§  423]  DENOMINATE   QUANTITIES  153 

2.   Reduce  -f$  of  a  Troy  pound    to  a  compound  denominate 
number. 

3 

J*_  y  Af  =  _  =  2  *  oz  SOLUTION.     The  successive  multipliers  are  12 

^014  and  20  respectively.     Multiplying  T3g  Ib.  by  12, 

4  the  result  is  2|  oz.     Multiplying  £  oz.  by  20,  the 

5  result  is  5  pwt.     Therefore,  ^  of  a  Troy  pound 
is  equal  to  2  oz,  6  pwt. 


S.  Eeduce  .3165  of  a  Troy  pound  to  a  compound  denominate 
number. 


SOLUTION.    The  successive  multipliers  are  12,  20,  and 
24  respectively. 

Multiplying  .3165  Ib.  by  12,  the  result  is  3.798  oz. 
Multiplying  .798  oz.  by  20,  the  result  is  15.96  pwt.     Multi- 
15.960  pwt.       plying  .96  pwt.  by  24,  the  result  is  23.04  gr.    Therefore, 
24  .3165  Ib.,  as  a  compound  denominate  number,  is  equal  to 

a  8  oz.  16  pwt.  23  gr. 


192 
23.04  gr. 


ORAL  EXERCISE 


1.  Reduce  ^  of  a  bushel  to  the  fraction  of  a  peck;  ^  of  a 
bushel;  -f$  of  a  bushel  to  a  compound  denominate  number. 

2.  What  part  of  a  shilling  is  \  of  -f^  of  a  pound  sterling  ? 

3.  What  decimal  of  an  inch  is  .08  of  a  foot  ?  .016  of  a  foot  ? 
4-  Reduce  ^  of  a  gallon  to  the  fraction  of  a  pint 

WRITTEN  EXERCISE 

1.  Reduce  -fa  of  a  Troy  pound  to  grains. 

2.  How  many  pennyweights  in  -fa  Ib.  ? 

S.  Reduce  f  of  a  mile  to  integers  of  lower  denominations. 

4.  Express  fj  of  an  acre  as  a  denominate  number. 

5.  Reduce  .1754  of  a  square  mile  to  lower  denominations. 

6.  Reduce       of  an  acre  to  lower  denominations. 


154 


DENOMINATE   NUMBERS 


[§§  424-427 


424.    Reduction  of  a  denominate  fraction  from  a  lower  denomination 
to  a  higher. 


425.    Example.     Reduce  f  of  a  grain  to  the  fraction  of  a  Troy 
pound. 

5     $1 
12 


SOLUTION.  Denominate  fractions 
may  De  reduced  from  a  lower  denomi- 
nation  to  a  higher  by  division  in  prac- 


20      12      14400 

tically  the  same  manner  as  denominate 
integers. 

Since  24  gr.  equal  1  pwt.,  20  pwt.  equal  1  oz.,  and  12  oz.  equal  1  lb.,  the 
successive  divisors  for  reducing  grains  to  pounds  are  24,  20,  and  12  respectively. 


ORAL  EXERCISE 

1.  What  part  of  a  foot  is  J  of  an  inch  ?   1  of  an  inch  ? 

2.  What  part  of  a  week  is  ^  of  a  day  ?   f  of  a  day  ? 
8.   What  part  of  a  gallon  is  ^  of  a  pint  ?   £  of  a  pint  ? 

4-   What  part  of  a  foot  is  .5  of  an  inch  ?   .025  of  an  inch  ? 


WRITTEN  EXERCISE 

1.  Reduce  %  of  a  penny  to  tho  fraction  of  a  pound  sterling. 

2.  Reduce  J  of  a  shilling  to  the  fraction  of  a  pound  sterling. 

3.  Change  2.4  cwt.  to  the  decimal  of  a  ton. 

Jf.   Reduce  -|  of  an  inch  to  the  decimal  of  a  yard. 

426.   Reduction    of   denominate    integers    to    fractions    of    higher 
denominations. 


427.   Examples. 

pound  sterling. 

(a) 
4 
12 
20 


3  far. 


9.5625s. 


£  .478125 


9s.  6d.  3  far.  =  459  far. 
£1=960  far. 
459  -r-  960  =  £  .478125 


1.   Reduce  9s.  6c2.  3  far.  to  the  decimal  of  a 


SOLUTIONS,  (a)  The  successive  divisors 
to  reduce  farthings  to  pounds  are  4,  12,  and 
20  respectively.  Dividing  3  far.  by  4,  the 
result  is  .75d  Putting  with  this  the  6d,  the 
result  is  6.75.  Dividing  6.76<Z.  by  12,  the  re- 
sult is  .5625s.  Putting  with  this  the  9s.,  the 
result  is  9.5625.  Dividing  by  20,  the  result  is 
.478125  pounds  sterling.  Or, 

(6)  In  9s.  6d.  3  far.  there  are  459  far.,  and 
in  £  1  there  are  960  far.  Hence,  9s.  6d.  3  far. 
is  ||§  of  a  pound  sterling.  £  ||jj  =  £  .478125. 


427-J28]  DENOMINATE   QUANTITIES  155 

2.   Reduce  4  yd.  2  ft.  6  in.  to  the  fraction  of  a  rod. 

SOLUTIONS,    (a)  The  successive  divisors 

to  reduce  inches  to  rods  are  12,  3,  and  6J 

6  -5-  12  =  J  ft.  respectively.    6  in.  divided  by  12  equal  £  ft. 

5  f9iN  _±_  3 5  vd  Putting  with  this  the  2  ft.,  the  result  is  2^  ft. 

2J  ft.  divided  by  3  equal  f  yd.    Putting  with 

V  (H)  •*•  5i  =  H  rd-  this  the  4.  yd.,  the  result  is  4|  yd.     4g  yd. 

divided  by  5£  equal  8|  rd.    Or, 

(d) 

4  yd.  2  ft.  6  in.  =  174  in.          (6)  4  yd.  2  ft.  6  in.  equal  174  in.    1  rd. 
1  VA   _  1  o«  ;«     equals  198  in.    4  yd.  2  ft.  6  in.  is  therefore  J$| 

JL      ICl.     —     J_«7O     111*  *    ^  ,  n  r\  •* 

of  1  rd.,  or  £$  rd. 


ORAL  EXERCISE 

1.  What  part  of  a  dollar  is  35^  ?  of  a  day  is  7  hr.  ? 

&  What  part  of  a  gallon  is  1  pt.  ?   of  a  rod  is  4  yd.  ? 

8.  What  decimal  of  a  bushel  is  1  pk.  ?   of  a  gallon  is  3  pt.  ? 

4.  What  decimal  of  a  yard  is  2  f t.  ?  of  £  1  is  12s.  ? 

WRITTEN  EXERCISE 

1.  Reduce  6  cwt.  54  Ib.  to  the  decimal  of  a  ton. 

&  What  decimal  of  an  acre  is  2722  sq.  ft.  72  sq.  in.  ? 

3.  What  fraction  of  a  pound  equals  11  oz.  11  pwt.  18  gr.  f 

4*  Reduce  3  oz.  11  pwt.  12  gr.  to  the  decimal  of  a  pound. 

6.  17  cwt.  72  Ib.  4  oz.  is  what  fraction  of  a  ton  ? 

ADDITION  OF  DENOMINATE  NUMBERS 

428.  Denominate  numbers  may  be  added,  subtracted,  multiplied, 
and  divided  upon  the  same  general  principles  by  which  similar 
operations  are  performed  in  simple  numbers.  The  only  variation 
arises  from  the  fact  that  in  simple  numbers  ten  units  of  any  lower 
denomination  make  one  of  the  next  higher,  while  in  denominate 
numbers  the  scale  is  not  at  all  uniform. 


156  DENOMINATE   NUMBERS  [§§  429-430 

429.  Example.     Find  the  sum  of  3  bu.  2  pk.  1  pt.,  5  bu.  1  qt., 
6  bu.  3  pk.  7  qt.  1  pt.,  2  bu.  1  pt. 

bu     pk    qt     pt  SOLUTION.     Write  the  numbers,  as  in  simple  addition, 

3201      so  that  tlie  units  °^  tne  same  denominations  stand  in  the 
1      A      same  vertical  column. 

Begin  at  the  right  and  add.    The  sum  of  the  pints  is 
637  3  pt.,  which  is  equal  to  1  qt.  1  pt.    Write  1  pt.  and  carry 

2001      Iqt. 
17     2     1     1  The  sum  of  the  quarts  is  9  qt.,  or  1  pk.  1  qt.    Write 

1  qt.  and  carry  1  pk. 

The  sum  of  the  pecks  is  6  pk.,  or  1  bu.  2  pk.    Write  2  pk.  and  carry  1  bu. 
The  sum  of  the  bushels  is  17,  which  write  as  bushels,  thus  completing  the 
addition. 

The  result  is  17  bu.  2  pk.  1  qt.  1  pt. 

WRITTEN  EXERCISE 

1.  What  is  the  sum  of  &  21  5s.  7d.  2  far.,  £  16  3s.  Sd.  1  far.,  £  14 
9s.  2  far.,  £  21  8s.  12d.  2  far.,  £  16  15s.  Id.  1  far.  ? 

2.  What  is  the  sum  of  1  mi.  8  rd.  5  yd.  2  ft.,  3  mi.  17  rd.  5  yd. 
2  ft.,  8  mi.  4  yd.  1  ft.,  4  mi.  1  yd.  1  ft.,  1  mi.  17  rd.  20  ft.  ? 

8.  What  is  the  sum  of  5  A.  110  sq.  rd.  5  sq.  ft.  28  sq.  in.,  1  A. 
80  sq.  rd.  3  sq.  ft.  12  sq.  in.,  12  A.  16  sq.  rd.  2  sq.  ft.  48  sq.  in.,  5  A. 
W  sq.  rd.  3  sq.  ft.  21  sq.  in.,  8  A.  100  sq.  rd.  3  sq.  ft.  42  sq.  in. 

4.  Add  436  Ib.  4  02. 15  pwt.,  83  Ib.  11  oz.  21  gr.,  46  Ib.  16  pwt., 
105  Jb.  9  oz.  11  gr. 

5.  What  is  the  sum  of  16  Ib.  16  pwt.  16  gr.,  100  Ib.  1  oz.  5  pwt 
20  gr.,  76  Ib.  7  oz.  5  pwt.  13  gr.,  19  Ib.  2  oz.  10  pwt.  20  gr.  ? 

SUBTRACTION  OP  DENOMINATE  NUMBERS 

430.  Example.     From  84  rd.  3  yd.  2  ft.  6  in.  take  12  rd.  5  yd. 
2  ft  8  in. 

rd.     yd.     ft.      in.  SOLUTION.    Write  the  numbers  so  that  the  units  of 

84     3       2       6  ^e  same  denominations  stand  in  the   same  vertical 

1 Q     K  column,  and  beginning  at  the  right  subtract  as  in  sim- 

ifLJL__f: °  pie  numbers. 

71     2\     2  10  Since  8  in.  cannot  be  subtracted  from  6  in.,  take  1  ft. 

|_1       Q  (12  in.)  from  2  ft.  and  add  it  to  6  in.,  making  18  in. 

£Z        £ I        ~.  18  in.  minus  8  in.  leaves  10  in.,  which  write  as  inches  in 

11       *     L  the  remainder. 


§§430-431]  DENOMINATE   QUANTITIES  157 

Inasmuch  as  1  ft.  was  added  to  6  in.,  there  is  but  1  ft.  remaining  in  the 
minuend.  Since  2  ft.  cannot  be  subtracted  from  1  ft.,  take  1  yd.  (3  ft.)  from 
the  3  yd.  and  add  it  to  the  1  ft.,  making  4  ft.  4  ft.  minus  2  ft.  leaves  2  ft., 
which  write  as  feet  in  the  remainder. 

Inasmuch  as  1  yd.  was  added  to  1  ft.,  there  are  but  2  yd.  remaining  in  the 
minuend.  Since  5  yd.  cannot  be  subtracted  from  2  yd.,  take  1  rd.  (5£  yd.)  from 
84  rd.  and  add  to  the  2  yd.,  making  1\  yd.  1\  yd.  minus  5  yd.  leaves  2£  yd., 
which  write  as  yards  in  the  remainder.  83  rd.  minus  12  rd.  leaves  71  rd.,  which 
write  as  rods  in  the  remainder. 

Reducing  J  yd.  to  lower  denominations  and  adding,  the  required  result 
is  found  to  be  71  rd.  3  yd,  1  ft.  4  in. 


WRITTEN  EXERCISE 

1.  From  a  barrel  containing  36  gal.  1  pt.  of  oil,  there  were  sold 
1  qt.  1  pt.  2  gi.  at  one  time,  and  at  another  21  gal.  2  qt.  1  pt.     How 
much  remained  unsold  ? 

2.  I  owned  640  A.  of  prairie  land.     After  selling  126  A.  45  sq.  rd. 
to  A,  and  117  A.  37  sq.  rd.  to  B,  how  much  had  I  left  ? 

3.  From  £  rd.  take  3J  ft. 

4.  Having  bought  21fJ  Ib.  of  old  silver,  I  used  15  Ib.  15  pwt. 
15  gr.     How  much  had  I  left  ? 

5.  Having  raised  214|  bu.  of  potatoes,  I  sold  125  bu.  3  pk. 
What  is  the  remainder  worth  at  50^  per  bushel  ? 

6.  Three  men  together  own  27  T.  75  Ib.  of  hay.     If  A  owns  9|  T., 
and  B  11  T.  75  Ib.,  how  much  does  C  own? 

7.  An  English  merchant  bought  goods  amounting  to  £  5926  12s. 
After  selling  a  part  of  the  goods  for  £4192  12s.  9d.,  he  took  an 
account  of  stock  and  found  that  he  had  on  hand  merchandise  worth, 
at  cost  prices,  £  2241  4s.  3d     Did  he  gain  or  lose,  and  how  much  ? 


Finding  the  Difference  between  Two  Dates 

431.  Difference  of  time  may  be  found  in  either  of  two  ways: 
(1)  by  compound  subtraction,  and  (2)  by  counting  the  actual  number 
of  days  from  the  given  to  the  required  date. 


158  DENOMINATE  NUMBERS  [§§432-433 

432.  In  all  business  transactions  involving  long  periods  of  time, 
the  difference  is  usually  found  by  compound  subtraction,  while  in 
transactions  involving  short  periods  of  time,  the  difference  is  usually 
found  by  counting  the  exact  number  of  days» 

433.  Examples.     1.   A  mortgage  dated  Sept.  22, 1892,  was  paid 
Aug.  31,  1903.     How  long  had  it  run  ? 

SOLUTION.    The  later  date  expresses  the  greater 

yr.  mo.  da. 

1903         8         31       leBgth  of  time  ;  hence,  write  it  as  the  minuend  and 
.„„  ~9       the  earlier  date  as  the  subtrahend.    August  being 

10MJ         y         AL       the  gth  montn  and  September  the  9th,  write  8  and  9 
10       11  9       respectively,  instead  of  the  names  of  the  months. 

Subtract  as  in  ordinary  denominate  numbers,  con- 
sidering 30  days  a  month  and  12  months  a  year.  As  near  as  the  time  can  be 
expressed  in  years,  months,  and  days,  the  mortgage  is  found  to  have  run  10  yr. 
11  mo.  9  da. 

2.  A  note  dated  June  7  was  paid  Sept  5.     How  many  days 
had  it  run? 

23  da.     in  June.  SOLUTION.    First  write  the  time 

o-.    -,       s-     -r  i  remaining  in  June,  then  the  actual 

number  of  days  in  July  and  August 

31  da.     in  August.  respectively,  and  finally  the  number 

5  da.     in  September.  of  days  in  September  up  to  and  in- 

90dS    from  Jvme  7  to  Sept.  5.      f ding  ^  5\  T^.8am  of  thes° 

days  is  the  required  time  expressed 

with  exactness.    Observe  that  the  aggregate  time  excludes  the  first  and  includoa 
the  last  day  of  the  dates. 

ORAL  EXERCISE 
By  inspection,  find  the  exact  number  of  days  between :  * 

1.  May  3  and  June  26.  9.   Mar.  20  and  Apr.  28. 

2.  June  25  and  Aug.  1.  10.  Apr.  3  and  June  1. 
8.  July  2  and  Aug.  31.  11.   Apr.  15  and  June  15. 

4.  Sept.  20  and  Oct.  31.  12.  June  2  and  Aug.  2. 

5.  Nov.  10  and  Dec.  31.  13.  Sept.  5  and  Nov.  8. 

6.  Oct.  16  and  Dec.  28.  14.  Oct.  15  and  Dec.  18. 

7.  Mar.  1  and  Apr.  10.  15.  Aug.  9  and  Oct.  21. 

8.  Jan.  9  and  Feb.  23.  16.  Sept.  19  and  Nov.  1. 

»  All  of  the  dates  given  are  in  the  same  year  unless  otherwise  specified. 


§§433-434]  DENOMINATE   QUANTITIES  159 

WRITTEN  EXERCISE 

Find  the  exact  number  of  days  between : 

1.  Aug.  19,  Nov.  23.  7.  May  29,  July  17. 

2.  May  3,  Dec.  25.  8.  May  1,  Dec.  28. 

S.  Sept.  29,  Aug.  18.  '  9.  Jan.  1, 1902,  Mar.  23,  1902. 

4.  Apr.  29,  July  25.  10.  Jan.  24,  1900,  Mar.  2,  1901. 

5.  Apr.  16,  Nov.  25.  11.  Sept.  30, 1903,  Mar.  6,  1905. 

6.  Mar.  18,  Nov.  6.  12.  July  27, 1900,  Dec.  27, 1902. 

By  compound  subtraction,  find  the  difference  between : 
IS.  Jan.  28, 1905,  Aug.  31, 1910.     17.  June  30,  1901,  Aug.  3, 1904. 

14.  Sept.  28, 1889,  Sept.  28, 1906.     18.   Sept.  18, 1896,  Feb.  11, 1904. 

15.  Mar.  5, 1899,  July  10,  1903.      19.   May  1, 1897,  June  2,  1900. 

16.  Feb.  11, 1900,  Jan.  6, 1903.       20.   May  25, 1883,  June  3, 1901. 

MULTIPLICATION  OF  DENOMINATE  NUMBERS 

434.   Example.    Multiply  3  bu.  1  pk.  2  qt.  1  pt.  by  12. 

bu.  pk.   qt.   pt.  SOLUTION.     Begin  at  the  right  and  multiply  as  in 

3121      simple  numbers. 

12  12  x  1  pt.  =  12  pt.  or  6  qt.    Write  0  in  the  place  of 


39     3     6     0      Pmts  in  the  product  and  carry  6  qt. 

12  x  2  qt.  =  24  qt.    24  qt.  -f  6  qt.  carried  from  pints 

equal  30  qt.,  or  8  pk.  6  qt.     Write  6  in  the  place  of  quarts  in  the  product  and 
carry  3. 

12  x  1  pk.  =  12  pk.    12  pk.  +  8  pk.  carried  are  15  pk.,  or  8  bu.  3  pk.    Write 
8  in  the  place  of  pecks  in  the  product  and  carry  8. 

12  x  3  bu.  =  36  bu.    36  bu.  -f  3  bu.  carried  are  39  bu.  which  write  in  the 
product.    The  complete  product  is  therefore  39  bu.  3  pk.  6  qt 


WRITTEN  EXERCISE 

1.  I  bought  7  Ib.  7  oz.  12  pwt.  18  gr.  of  old  gold  at  $1.05  per 
pennyweight.     What  was  the  value  ? 

2.  What  is  the  cost  of  15  chains  of  gold  each  weighing  3  oz. 
17  pwt.  17  gr.  at  7^  per  grain? 


160  DENOMINATE   NUMBERS  [§§  434-436 

8.  How  much  hay  in  9  stacks  each  containing  5  T.  21  cwt. 
83  Ib.  ?  What  is  the  hay  worth  at  $  12.75  per  ton  ? 

4-  A  merchant  bought  40  Ib.  8  oz.  of  sugar  at  6J^  per  pound, 
and  5  times  as  much  coffee  at  33^  per  pound.  What  was  the 
amount  of  his  bill  ? 

5.  If   a  piece   of  land   produces    on   an   average    133J  bu.  ol 
potatoes  per  acre,  what  will  be  the  value  of  6  A.  80  sq.  rd.  at  66|  ^ 
per  bushel?   at  75^  per  bushel  ?   at  50^  per  bushel  ? 

6.  A  man  owning  a  farm  of  400  A.  of  land  sold  125  A.  75  sq.  rd. 
How  much  is  the  remainder  of  the  farm  worth  at  $  75  per  acre  ? 

7.  A  man  bought  30  piles  of  wood,  each  containing  4  cd.  80  cu.  ft. 
at  $3.33J  per  cord.     He  later  sold  the  entire  quantity  for  $560. 
Did  he  gain  or  lose,  and  how  much  ? 

8.  Keduce  £  15  3s.  Id.  2  far.  to  United  States  money. 

SOLUTION.  Since  £1  equals  $4.8665,  £15  equal  $  72.9975 ;  since  Is. equals 
24^,  3s.  equal  73^ ;  since  Id.  equals  2^^,  Id.  equal  $.1414  ;  since  1  far.  equals 
§§p,  2  far.  equal  $.0101.  Add  these  separate  equivalents  together,  and  the  result 
is  $73.879,  or  $73. 88.  Or, 

Call  each  2  shillings  ^  of  a  pound,  then  3  shillings  equals  £.15;  call  the 
pence  and  farthings,  reduced  to  farthings,  so  many  YflW  of  a  pound.  7  pence 
plus  2  farthings  equals  30  farthings,  equals  £.030;  to  this  add  the  £15  and  the 
£.15,  and  the  result  is  £15.18.  Since  £1  equals  $4.8665,  £15.18  equal  15.18 
times  $4.8665,  or  73.87347  =  $73.87. 

9.  Keduce  £25  4s.  Sd.  to  United  States  money. 


DIVISION  OP  DENOMINATE  NUMBERS 

435.   Example.    If  15  yd.  of  cloth  are  worth  £23  11s.  3d,  what 
is  1  yd.  worth  ? 

£        «.       d.  SOLUTION.     The  cost  of  1  yd.  is  ^  of  the  cost  of 

15)23      11      3      15  yd.     ^  of  £23  is  £1  with  an  undivided  remainder 

1      11      5       of  £8.    Write  £  1  in  the  quotient  and  add  the  remainder 

to  the  next  lower  denomination.     £8  lls.  =  171s.     ^ 

of  171s.  is  lls.  with  an  undivided  remainder  of  6s.  Write  11s.  in  the  quotient 
and  add  the  remainder  to  the  next  lower  denomination.  6s.  3d.  =  75d.  ^3  of 
75d.  is  5d,  which  write  in  the  quotient.  1  yd.  of  cloth  is  therefore  worth  £  1 
lls.  5d. 


§435]  DENOMINATE  QUANTITIES  161 

WRITTEN  EXERCISE 

Find  the  quotient  of: 

1.  24  bu.  3  pk.  -h 9.  8.  £28  10s.  6d.  2  far.  -*-6. 

&  16  T.  9cwt.  241b.-*-16.       4.  25  yr.  5  da.  12  hr.  -f-  8. 

5.  If  7  Ib.  7  oz.  12  pwt.  18  gr.  of  silver  be  made  into  6  plates  of 
equal  weight,  what  will  be  the  weight  of  each  ? 

6.  A  miner  having  63  Ib.  1  oz.  10  pwt.  of  gold  dust  divided  \  oi 
it  among  his  laborers,  and  had  the  remainder  made  into  chains 
averaging  3  oz.  3  pwt.  3  gr.  of  pure  gold  each.    If  he  sold  the  chains 
for  $  72.50  each,  how  much  did  he  receive  for  them  ? 

7.  From  the  sum  of  £25  4s.  lOrf.  and  £10  5.s.  2d.  take  theii 
difference,  divide  the  result  by  5,  and  reduce  the  quotient  to  United 
States  money. 

8.  I  sold  98  cd.  96  cu.  ft.  of  wood  for  $395,  and  in  so  doing  lost 
$  98.37|.    What  did  the  wood  cost  per  cord  ? 

9.  Reduce  $5164.28  to  equivalent  English  money. 

SOLUTION 

$5164.28  H-  4.8665  =  £1.061.189. 

£.189  x  20  =  3.78«. 

.78*  X  12  =  9.36d. 

.86  d.  x4  =  1.44far. 

Hence,  $5164.28  =  £1061  3s.  Qd.  I  far. 

10.  Reduce  $185  to  equivalents  in  English  money  j  $2500. 

WRITTEN  REVIEW 

1.  How  many  fields,  each  of  10  A.  56  sq.  rd.  21  sq.  yd.  5  sq.  ft. 
and  28  sq.  in.,  can  be  formed  from  a  farm  containing  124  A.  40  sq.  rd. 
16  sq.  yd.  8  sq.  ft.  48  sq.  in.? 

2.  Reduce  $3750  to  English  money. 

S.  To  the  sum  of  -J,  \ ,  and  ^  of  an  acre,  add  .0055  of  a  square 
mile. 

4.  From  .6375  of  an  acre  take  yf  of  a  square  rod. 

5.  A  wheelman  ran  71  mi.  246  rd.  1  yd.  2  ft.  6  in0  in  the  fore- 
noon, and  20  mi.  10  rd.  8  in.  less  in  the  afternoon.     What  distance 
did  he  run  in  the  entire  day  ? 


162  DENOMINATE   NUMBERS  [§  435 

6.  A.  grocer  bought  110  qt.  of  chestnuts  by  dry  measure,  and 
when  selling  them  used  a  liquid  pint  measure.     What  was  his  gain 
if  he  bought  the  chestnuts  at  6^  a  quart  and  sold  them  at  5^  a  pint  ? 

SOLUTION 

1  dry  quart  =  67£  cu.  in. 

1  liquid  quart  =  57 f  cu.  in. 

67£  cu.  in.  x  110  =  7392  cu.  in. 

7392  cu.  in.  -4-  57f  cu.  in.  =  128. 

1  qt.  x  128  =  128  qt.,  or  256  pt. 

110  qt.  at  6^  =  $6.60,  the  cost. 

256  pt.  at  5^  =  $12.80,  the  selling  price. 

$12.80  -  $6.60  =  $6.20,  the  gain. 

7.  A  blundering  clerk  bought  of  a  gardener  192  qt.  of  currants, 
measuring  them  by  a  liquid  quart  measure  and  selling  them  by  dry 
measure.    If  the  currants  were  bought  at  6^  per  quart  and  sold  at  7^, 
did  he  gain  or  lose,  and  how  much  ? 

8.  From  a  cask  of  brandy  containing  69  gal.  1  pt.,  and  costing 
$3.75  per  gallon,  \  leaked  out  and  the  remainder  was  sold  at  20^  per 
gill.    What  was  the  amount  of  the  gain  or  loss  ? 

9.  If  2  qt.  1  pt.  1  gi.  of  oil  be  consumed  per  day  for  the  year 
1903,  what  will  be  its  cost  for  the  year  at  8^  per  gallon  ? 

10.  I  bought  by  Avoirdupois  weight  28T8Tr  Ib.  of  drugs  and  from 
the  stock  sold  by  Apothecaries'  weight  29  Ib.    What  is  the  remainder 
worth  at  75^  per  Apothecaries'  ounce  ? 

11.  A  farmer  sold  4  loads  of  hay,  weighing  respectively  1  T. 
2  cwt.  14  Ib.,  19  cwt.  90  Ib.,  1  T.  3  cwt.  97  Ib.,  1  T.  5  cwt.,  and 
received  for  it  $  16  per  ton.     How  much  did  he  receive  ? 

12.  A  goldsmith  bought  3  Ib.  9  oz.  1  pwt.  16  gr.  of  old  gold  at 
80^  per  pennyweight,  and  made  it  into  pins  of  40  gr.  weight  each, 
which  he  sold  at  $  2  apiece.     How  much  did  he  gain  or  lose  ? 

18.  73,920  qt.  of  cherries  were  bought  at  10^  per  quart  dry  meas- 
ure, and  sold  at  the  same  price  liquid  measure.  How  much  was 
thereby  gained  ? 

14-  A  horse  requires  ^  of  a  bushel  of  oats  per  day.  At  9^  a 
peck,  how  much  will  it  cost  to  feed  him  oats  for  July  and  August  ? 


§§  436-442] 


PRACTICAL   MEASUREMENTS 


163 


PRACTICAL  MEASUREMENTS 
DISTANCES 

436.  An  angle  is  the  difference  in  direction  of  two  lines  proceed- 
ing from  a  common  point  called  the  vertex. 

A 

437.  A  right  angle  is  an  angle  formed  when 
one  straight  line  meets  another  so  as  to  make 
the  adjacent  angles  equal.     The  lines  forming 
the  angles  are  said  to  be  perpendicular  to  each 
other. 


In  the  accompanying  diagram  ABC  and  ABD  are 
right  angles,  and  the  lines  AB  and  CD  are  perpendicu- 
lar to  each  other. 

438.  A  triangle  is  a  plane  figure  with  three 
plane  sides  and  three  plane  angles.     The  side 
on  which  the  triangle  stands  is  the  base,  the 
opposite  corner  the   vertex,  and  the   shortest 
distance  from  the  vertex  to  the  base,  or  the 
base  extended,  is  the  height  or  altitude  of  the 
triangle. 

439.  A  right-angled  triangle  is  a  triangle 

having  a  right  angle. 

440.  A  rectangle  is  a  plane  figure  having 
four  straight  sides  and   four  square   corners. 
When  the  four  sides  are  equal,  the  figure  is 
usually  called  a  square. 

441.  The  perimeter  of  any  plane  figure  is 
the  length  of  the  line  or  lines  inclosing  it. 

442.  When  a  plane  figure  is  bounded  by  a 
curved  line,  every  point  of  which  is  equally  dis- 
tant from  the  center,  it  is  called  a  circle.     The 
perimeter  of  a  circle  is  called  its  circumference; 
a  line  passing  through  the  center  and  terminat- 
ing in  the  circumference,  the   diameter;  one 
half  of  the  diameter,  the  radius. 


Right 
Angle 


Right 
Angle 


D 


Triangle 


Right-angled  Triangle 


Rectangle 


164  DENOMINATE   NUMBERS  [§§  443-444 

443.  To  find  the  circumference  of  a  circle  when  the  diameter  is  given, 
Multiply  the  diameter  by  3.1416. 

444.  To  find  the  diameter  of  a  circle  when  the  circumference  is  given, 
Divide  the  circumference  by  3.1416. 

ORAL  EXERCISE 

1.  How  many  rods  of  fence  will  inclose  a  farm  in  the  form  of  an 
equilateral  triangle  each  side  of  which  is  25  rd.  ? 

2.  How  many  rods  of  fence  will  inclose  a  field  15  rd.  long  and 
10  rd.  wide  ? 

3.  It  required  100  rd.  to  inclose  a  garden  which  is  in  the  form  of 
a  perfect  square.     How  many  feet  in  each  side  of  the  garden  ? 

4.  What  is  the  radius  of  a  circle  whose  circumference  is  314.16  ft.  ? 

5.  How  many  yards  of  border  will  be  required  for  the  paper  in  a 
room,  the  length  and  width  of  whose  sides  are  18  ft.  and  12  ft.  respec- 
tively? 

6.  How  many  posts  16  J  ft.  apart  will  be  required  for  the  fence  of 
a  square  field  whose  sides  are  each  100  rd.  ? 

7.  The  perimeter  of  a  rectangle  is  72  rd.     If  the  width  is  12  rd., 
what  is  the  length  ? 

8.  If  it  cost  $  75  to  fence  a  certain  square  field,  how  much  would 
it  cost  to  fence  a  similar  field  whose  sides  are  double  the  length  of 
those  of  the  first  field  ? 

WRITTEN  EXERCISE 

1.  What  is  the  circumference  of  a  circle  whose  diameter  is  24  ft.  ? 

2.  What  will  be  the  cost  of  the  posts  required  to  fence  a  square 
field  whose  sides  are  each  16,500  ft.,  if  the  posts  are  set  1  rd.  apart 
and  cost  $  8.37^  per  C? 

3.  Find  the  cost  of  the  wire  necessary  to  fence  a  rectangular  field 
whose  length  and  breadth  are  24  rd.  and  18  rd.  respectively,  if  the 
fence  contains  five  wires  and  the  wire  is  worth  \  $  per  foot. 

4.  How  many  feet  of  fence  will  inclose  a  circular  field  8.5  rd.  in 
diameter  ? 

5.  What  is  the  radius  of  a  circle  whose  circumference  is  7854  ft.  ? 

6.  A  room  is  in  the  form  of  a  rectangle  471  ft.  long  and  36|  ft. 
wide.     How  many  feet  and  inches  of  molding  will  be  required  for 
its  four  walls  ? 


§§  445-447] 


PRACTICAL  MEASUREMENTS 


166 


AREAS 

445.  The  dimensions  of  a  surface  are  length  and  breadth  (364). 
In  finding  the  area  of  any  surface  having  a  uniform  length  and 
breadth  it  is  necessary  to  select  a  measuring  unit.     This  may  be  any 
square,  each  side  of  which  is  a  unit  of  length  (370) .     The  number 
of  square  units  found  in  the  surface  is  the  required  area. 

To  illustrate,  take  the  following  example. 

446.  Example.    What  is  the  area  of  a  garden  5  rd.  long  by  66  ft. 
wide? 

SOLUTION.  Select  a  measuring  unit.  This  may  be  any  square,  each  side  of 
which  is  a  unit  of  length. 

Since  66  ft.  is  equal  to  just  4  rd.,  take  1  sq.  rd.  (a)  as  a  measuring  unit. 
Then,  if  lines  are  drawn  as  represented  in  (c),  the  entire  surface  will  be  divided 

(a)  (c) 

1  rd.  5  rd. 


Ird. 


(6) 

5rd. 

Ird. 

5 

sq.  rd. 

2C 


mto  square  rods.  In  the  horizontal  row  (5)  there  will  be  5  sq.  rd.,  and  since  in 
che  entire  figure  there  will  be  4  horizontal  rows  which  are  equal  to  (6),  the 
area  is  4  times  5  sq.  rd.,  or  20  sq.  rd.  Hence, 

The  dimensions  of  a  rectangle  reduced  to  units  of  the  same  denomi- 
nation and  multiplied  together  express  the  area  of  a  rectangle  in  square 
units  having  the  same  denomination  as  the  units  of  length. 

447.    Example.    Find  the  area  of  a  triangle  whose 
tude  are  6  and  8  ft.  respectively. 

SOLTTTIOW.  In  the  accompanying  diagram  assume  that 
the  base  (GB)  is  6  ft.  and  the  altitude  (^l-D)  is  8  ft.  It  will 
be  seen  that  the  altitude  divides  the  triangle  into  two  right- 
angled  triangles,  each  of  which  is  one  half  of  a  rectangle 
whose  sides  are  8  ft.  and  8  ft.  Two  triangles,  each  one 
half  of  a  rectangle  8  ft.  by  3  ft.,  are  equal  to  one  rectangle 
8  ft.  by  3  ft.  The  area  of  the  triangle  given  is,  then,  the 
product  of  these  two  dimensions,  or  24  sq.  ft.  Hence, 


166  DENOMINATE   NUMBERS  [§§448-449 

448.  To  find  the  area  of  a  triangle,  the  base  and  altitude  being  given, 
Multiply  the  altitude  by  one  half  the  base. 

449.  To  find  the  area  of  a  circle,  the  circumference  and  diameter 
being  given, 

Square  the  r&dius  and  multiply  by  3.1416  ;  or  square  the  diameter 
and  multiply  by  .7854  (J  of  3.1416). 

ORAL  EXERCISE 

1.  How  many  square  feet  in  the  ceiling  of  a  room  25  ft.  long 
and  16  ft.  wide  ? 

2.  What  is  the  width  of  a  rectangle  12  ft.  long  if  it  contains 
the  same  area  as  a  square  8  ft.  on  a  side  ? 

3.  What  is  the  area  of  a  triangle  whose  base  is  11  ft.  and  whose 
altitude  is  15|  ft.  ? 

4.  A  room  15  ft.  long  and  8  ft.  wide  has  a  tile  floor.     If  one  tile 
is  4  in.  square,  how  many  tiles  in  the  floor  ? 

5.  What  is  the  difference  between  a  square  rod  and  a  rod  square  ? 
between  9  sq.  ft.  and  9  ft.  sq.  ?  between  ^  of  a  square  foot  and  1  of 
a  foot  square  ? 

6.  Express  in  inches  the  difference  between  J  of  a  square  foot 
and  J  of  a  foot  square. 

WRITTEN   EXERCISE 

Find  the  number  of  acres  in  rectangular  fields  containing  dimen- 
sions as  follows.    Perform  the  multiplications  as  explained  in  91-93. 

1.  36rd.by21rd.        4.  72  rd.  by  23  rd.          7.  47  ch.  by  95  ch. 

2.  62rd.by51rd.        5.  75  rd.  by  32  rd.         8.  51  ch.  by  49  ch. 

3.  85rd.by64rd.        6.  84  rd.  by  23  rd.          9.  75  ch.  by  52  ch. 

10.  A  field  87£  rd.  wide  and  240  rd.  long  produced  27|  bu.  of 
wheat  to  the  acre.     What  was  the  crop  worth,  at  90  ^  per  bushel  ? 

11.  A  farm  in  the  form  of  a  rectangle  is  75  rd.  wide ;  if  the  area 
is  167.5  A.,  how  long  is  the  farm  ? 


§§449-452]  PRACTICAL   MEASUREMENTS  167 

12.    I  wish  to  build  a  shed  which  will  cover  £  of  an  acre  of  land. 

O 

If  the  width  of  the  shed  is  42  ft.,  what  must  be  its  length  ? 

18.    17.75  bu.  of  timothy  seed  is  sown  on  land  at  the  rate  of  6  Ib. 
per  acre.     What  will  be  the  area  of  the  land  thus  seeded  ? 

14.  A  hall  1\  ft.  wide  and  19|  ft.  long  is  covered  with  oilcloth, 
at  65  $  per  square  yard.     How  much  did  it  cost  ? 

15.  A  city  lot  in  the  shape  of  a  triangle  has  a  base  of  90  yd.  and 
an  altitude  of  120  yd.     What  is  it  worth  at  $2.50  per  square  foot? 

16.  What  is  the  area  of  a  semicircle  if  the  radius  of  the  whole 
circle  is  100  ft.  ? 

17.  It  cost  $25  to  carpet  a  square  room  20  ft.  on  a  side.     At 
that  rate,  what  will  it  cost  to  carpet  a  square  room  40  ft.  on  a 
side? 

18.  A  rectangular  field  containing  54  A.  is  30  rd.  wide.     What 
will  it  cost  to  fence  it  at  2      a  foot  ? 


CARPETING 

450.  Carpet  is  sold  by  the  yard.      Oilcloth  and  linoleum  are 
sometimes  sold  by  the  square  yard. 

451.  In  finding  the  number  of  yards  of  carpet  for  a  room  it  is 
always  necessary  to  know  whether  the  strips  are  to  run  lengthwise 
or  crosswise. 

Merchants  will  sell  fractional  lengths,  but  not  fractional  widths 
of  carpet  ;  hence,  in  finding  the  cost  of  carpet  the  number  of  w.hole 
strips  must  be  found  and  1  added  to  this  number  for  all  fractions. 

452.  Where  the  length  of  the  strip  is  not  an  even  number  of 
yards,  there  is  usually  some  waste  in  matching  the  figures  of  the 
pattern,  and  since  dealers  charge  for  the  goods  furnished,  regardless 
of  the  waste,  all  items  of  wastage  must  be  included  in  the  cost. 

Often  carpet  may  be  laid  with  less  waste  one  way  of  the  room  than  another  ; 
hence,  in  estimating  the  cost  of  carpet  it  is  sometimes  desirable  to  find  the  cost 
of  the  strips  running  both  lengthwise  and  across  the  room,  and  then  to  make 
a  comparison. 


168  DENOMINATE   NUMBERS  [§453 

453.  Example.  How  many  yards  of  Brussels  carpet  f  of  a  yard 
wide,  laid  lengthwise  of  the  room,  will  be  required  to  cover  a  floor 
22  ft.  by  17  ft.  4  in.,  if  the  waste  in  matching  be  6  in.  on  each  strip  ? 

17  ft  4  in  =  42-  ft  =  5-£  yd  SOLUTION.     Since  the  strips  run  length. 

42  vd  ^ 3*_ si xk _ Loi -1 71 9  wise  °f  the  r°°m> to find  the number °f 

9    Ju    •  t  —  "T  A  3  —  27   -       2T«  strips,  divide  the  width  of  the  room  by 
7#  strips  is  practically  8  strips.   the  width  of  the  carpet    ¥  yd>  divided 

22  ft.  +  6  in.  =  22  J  ft.  by  £  yd.  equals  7£f ,  the  number  of  times 

22-J-  f t.  X  8  =  180  ft.  =  60  yd.          one  strip  required.  Since  fractional  widths 

of  carpet  cannot  be  bought,  drop  the  frac- 
tion and  add  1  to  the  whole  number ;  8  strips  are  required,  and  ^  of  a  strip 
may  be  cut  off  or  turned  under.  The  length  of  the  room  is  22  ft.  and  there  is 
a  waste  of  6  in.  on  each  strip  for  matching ;  hence,  the  length  to  be  bought  for 
each  strip  is  22$  ft.  If  there  are  22$  ft.  in  each  strip,  in  the  8  strips  there  are  8 
times  22$  ft.  or  180  ft.,  or  60  yd. 


WRITTEN  EXERCISE 

1.  How  many  yards  of  carpet,  1  yd.  wide,  laid  lengthwise  of 
the  room,  will  be  required  to  cover  a  floor  10.5  yd.  long  by  6  yd. 
wide,  if  no  allowance  is  to  be  made  for  matching  ? 

2.  What  will  be  the  cost  of  the  carpet  border  for  a  room  16J  ft. 
by  21  ft.,  if  the  price  be  62£^  per  yard  ? 

S.  How  many  yards  of  carpeting  f  yd.  wide  will  be  required  to 
carpet  a  room  32  ft.  long  by  25  ft.  wide,  if  the  lengths  of  carpet  are 
laid  across  the  room  and  8  in.  are  lost  on  each  strip  in  matching  the 
pattern  ?  How  many  yards  if  the  strips  are  laid  lengthwise  and  6  in. 
are  lost  in  matching  ?  If  the  carpet  is  laid  in  the  most  economical 
way,  what  will  be  the  cost  at  $  2.55  per  yard  ? 

4.  How  many  yards  of  Axminster  carpeting  f  of  a  yard  in  width, 
and  laid  lengthwise  of  the  room,  will  be  required  to  cover  a  floor 
21|  ft.  long  and  18|  ft.  wide,  making  no  allowance  for  waste  in 
matching  the  design  ? 

5.  Find  the  cost  of  a  carpet  f  yd.  wide,  at  $  2.50  per  yard,  for  a 
room  22  ft.  by  18  ft.,  if  the  strips  run  lengthwise  and  there  is  a  waste 
of  J  of  a  yard  on  each  strip  for  matching  the  pattern. 

6.  How  many  yards  of  ingrain  carpeting  1  yd.  wide  will  be 
required  to  cover,  lengthwise,  the  floor  of  a  room  26  ft.  long  by  19  ft. 


§§453-456]  PRACTICAL   MEASUREMENTS  169 

6  in.  wide,  the  waste  being  6  in.  on  each  strip  ?  How  many  yards  of 
Brussels  carpeting  27  in.  wide  will  cover  the  same  room  if  there  is  a 
waste  of  18  in.  per  strip  ? 

7.  What  will  it  cost,  at  $  1.15  per  yard,  to  carpet  a  flight  of  stairs 
11  ft.  4  in.  high,  the  tread  of  each  stair  being  10  in.  and  the  risei 
8  in.? 

8.  A  hall  8^  ft.  wide  and  18|  ft.  long  was  covered  with  oilcloth 
at  75  ^  per  square  yard.     How  much  did  it  cost  ? 

PAPERING 

454.  Wall  paper  is  usually  18  inches  wide,  and  may  be  bought  iiy 
single  rolls  8  yards  long  or  in  double  rolls  16  yards  long. 

455.  These  dimensions  are  for  the  wall  paper  commonly  used  in 
America.  *  Imported  papers  and  some  American  papers  vary  in  width 
and  length. 

In  practice  there  is  usually  considerable  waste  in  cutting  and 
matching  the  paper,  and  it  is  found  more  economical  to  buy  double 
rolls. 

There  is  no  uniform  rule  respecting  the  allowance  to  be  made  for  openings, 
such  as  doors  and  windows.  Generally  paper  hangers  estimate  the  number  of 
full  strips  that  would  be  necessary  for  the  regular  surface  of  the  walls,  and 
divide  this  number  by  the  whole  number  of  strips  that  can  be  cut  from  a  full  roll 
of  paper.  By  this  method  the  ends  of  the  rolls  are  supposed  to  be  utilized  for 
the  surface  above  doors  and  above  and  below  windows,  and  other  similar  irregu- 
lar places. 

456.  It  is  hardly  possible  to  determine  in  advance  the  exact 
number  of  rolls  of  paper  required  for  the  walls  of  any  room,  but  for 
practical  purposes  the  following  method  will  be  found  to  approxi- 
mate accuracy : 

From  the  perimeter  of  the  room  deduct  the  width  of  the  doors  and 
windows. 

Find  the  number  of  strips  necessary  for  the  regular  surface  of  the 
walls,  and  divide  the  result  by  the  whole  number  of  strips  that  can  be 
cut  from  a  full  roll  of  paper. 

The  quotient  will  be  the  number  of  rolls  required. 

For  standard  rolls  there  will  be  twice  as  many  strips  of  paper  required  as 
there  are  yards  in  the  length  of  the  regular  surface  to  be  covered. 

Any  whole  rolls  left  over  after  papering  may  be  returned  to  the  seller,  but 
no  portion  of  a  roll  will  ever  be  taken  back. 


170  DENOMINATE   NUMBERS  [§§  457-458 

457.  Example.  How  many  double  rolls  of  paper  will  be  required 
for  the  sides  and  ends  of  a  room  24  ft.  long,  18  ft.  wide,  and  8  ft. 
high,  with  1  door  and  3  windows,  each  3J  ft.  wide,  making  no  allow- 
ance for  waste  in  cutting  ? 

SOLUTION. 


24  ft.  +  18  ft.  x  2  =  84  ft.,  the  perimeter  of  the  room. 

3|  ft.  x  4  =  14  ft.,  the  width  of  the  doors  and  windows. 

84  ft.  -  14  ft.  =  70  ft.,  or  23£  yd.,  the  length  of  the  regular  surface  of  the 
walls. 

A  double  roll  of  paper  is  £  yd.  wide  and  48  ft.  long. 

23£  -=-  £  =  46|,  or  practically  47,  the  number  of  strips  necessary  for  the 
regular  surface. 

48  ft.  -=-  8  f t.  =  6,  the  number  of  strips  in  each  double  roll. 

47  -r-  6  =  7 1,  or  practically  8  double  rolls. 

Hence,  8  is  the  required  number  of  double  rolls  of  paper. 

*• 

WRITTEN  EXERCISE 

1.  How  many  single  rolls  of  paper  8  yd.  long  and  18  in.  wide  will 
it  take  to  cover  the  ceiling  of  a  room  60  ft.  long,  45  ft.  wide,  if  there 
be  no  waste  in  matching  ? 

2.  What  is  the  cost  of  paper,  at  $  1.25  per  double  roll,  for  a  room 
18  ft.  long,  12  ft.  wide,  and  9  ft.  high  above  the  baseboard,  allowing 
for  1  door  and  2  windows,  each  3J  ft.  wide  ? 

3.  How  many  rolls  of  paper  8  yd,  long  and  18  in.  wide  will  be 
required  for  the  walls  of  a  room  20  ft.  long,  15  ft.  wide,  and  having 
a  height  of  8  ft.  9  in.,  allowing  for  1  door  3  ft.  by  7  ft.,  and  for  2 
windows  3  ft.  by  6  ft.,  and  a  baseboard  9  in.  high  ? 

4-  At  $  1.90  per  double  roll,  what  will  be  the  cost  of  papering  a 
parlor  20  ft.  square  and  8  ft.  high  from  the  baseboard,  allowing  for 
1  door  3  ft.  by  7  ft.  and  3  windows,  each  3  ft.  by  6  ft.  ? 

5.  Allowing  for  3  windows  each  42  in.  by  7  ft.,  and  2  doors  each 
4  ft.  by  11  ft.,  what  will  be  the  cost,  at  $1.80  per  single  roll,  of 
papering  a  room  24  ft.  long,  18  ft.  wide,  and  12  ft.  high  from  the 
baseboard  ? 

PAINTING  AND  PLASTERING 


The  unit  of  painting  and  plastering  is  the  square  yard. 

Allowances  are  frequently  made  for  one  half  or  the  whole  of  the  area  of  the 
openings  and  the  baseboard;  but  since  there  is  no  uniform  custom  governing 
such  allowances,  a  written  contract  definitely  referring  to  this  matter  should  be 
drawn  up  ;  then  complications  at  the  time  of  settlement  will  be  avoided. 


§§458-461]  PRACTICAL   MEASUREMENTS  171 

WRITTEN  EXERCISE 

1.  At  11  /  per  square  yard,  what  will  be  the  cost  to  plaster  the 
sides  and  ceiling  of  a  room  that  is  30  ft.  long,  24  ft.  wide,  and  14  ft. 
high  above  the  baseboard,  making  full  allowance  for  4  doors  3  ft. 
6  in.  by  8  ft.  3  in.,  and  6  windows  3  ft.  8  in.  by  7  ft.  ? 

2.  At  20  ^  per  square  yard,  what  will  it  cost  to  paint  a  floor  40  ft. 
long  and  26  ft.  wide  ? 

3.  At  22^  per  square  yard,  what  will  it  cost  to  plaster  the  sides 
and  ceiling  of  a  room  30  ft.  by  24  ft.  by  12  ft.,  if  J-  of  the  surface  of 
the  sides  is  allowed  for  the  doors,  windows,  and  baseboard  ? 

4.  What  will  be  the  cost  at  18^  per  square  yard  for  plastering 
the  ceiling  and  walls  of  a  room  60  ft.  long,  30  ft.  wide,  and  15  ft. 
high  above  the  baseboard,  allowance  being  made  for  6  doors  4  ft. 
6  in.  wide  by  10  ft.  6  in.  high  above  the  baseboard,  and  12  windows 
each  3  ft.  6  in.  wide  by  8  ft.  high  ? 

5.  At  21  ^  per  square  yard,  what  will  it  cost  to  paint  both  sides 
of  a  board  partition  90  ft.  long  and  9  ft.  3  in.  high  ? 

6.  Allowing  J  of  the  surface  of  the  sides  for  doors,  windows,  and 
baseboard,  what  will  it  cost  at  12J^  per  square  yard  to  plaster  the 
sides  and  ceiling  of  a  room  30  ft.  long,  18  ft.  wide,  and  15  ft.  high  ? 

ROOFING  AND  FLOORING 

459.  The  unit  used  in  determining  the  number  of  square  feet  in 
any  roof  or  floor  is  a  square  10  ft.  on  a  side,  or  100  sq.  ft. 

460.  Shingles  are  16  in.  long  and  on  an  average  4  in.  wide.     They 
are  usually  laid  about  4^-  in.  to  the  weather,  1  shingle  covering  18  sq.  in. 
of  roof.     At  this  rate  it  requires  8  shingles  for  each  square  foot  of 
roof,  or  800  for  each  square  of  100  sq.  ft.     Allowing  for  the  waste 
that  is  usual  in  shingling,  about  1000  shingles  are  estimated  for  each 
square  of  100  sq.  ft.     Some  shingles  run  better  than  this,  and  from 
850  to  900  are  regarded  as  an  ample  number  for  a  square. 

461.  A  bundle  contains  250  shingles ;  hence  it  usually  requires 
about  4  bundles  or  1000  shingles  for  100  sq.  ft.  of  roof. 


172  DENOMINATE  NUMBERS  [§§461-464 

WRITTEN  EXERCISE 

1.  I  wish  to  floor  and  ceil  a  room  25  yd.  long,  15  yd.  wide.    What 
will  be  the  cost  of  the  material  at  $  27  per  thousand  square  feet  ? 

2.  Find  the  cost  at  $45  per  thousand  square  feet  of  the  flooring 
for  a  room  40  ft.  by  30  ft.,  the  waste  being  i  of  the  area  of  the  floor. 

3.  Find  the  cost  of  laying  an  oak  floor  which  is  18  ft.  by  16-  ft., 
if  the  labor  and  other  incidentals  amount  to  $25,  the  price  of  the 
lumber  is  $75  per  thousand  sq.  ft.,  and  an  allowance  of  42  sq.  ft. 
is  made  for  waste. 

4.  Counting  1000  shingles  for  120  sq.  ft.,  how  many  will  be 
required  to  cover  the  pitched  roof  of  a  barn  120  ft.  long  and  30  ft. 
wide  on  each  side  ? 

5.  Allowing  800  shingles  to  a  square,  how  many  thousand  will  be 
required  for  the  roof  of  a  barn  30  ft.  wide  on  each  side  and  100  ft. 
long?     What  will  be  the  cost  at  $3.50  per  thousand? 

6.  At  $  10  per  square,  what  will  be  the  cost  of  the  slate  for  a  roof 
60  ft.  long  and  32  ft.  wide  ? 

SOLID  CONTENTS 

462.  A  rectangular  solid  is  a  solid  bounded  by  six  rectangular 
sides  or  faces.     When  these  sides  are  squares,  the  figure  is  called  a 
cube. 

463.  The  dimensions  of  solids  are  length,  breadth,  and  thickness 
(365).     In  finding  the  solid  contents  of  any  solid  having  a  uniform 
length,  breadth,  and  thickness,  it  is  necessary  to  select  a  measuring 
unit.     This  may  be  any  cube,  each  side  of  which  is  a  unit  of  length 
(382).     The  number  of  cubic  units  found  in  the  solid  is  the  required 
solid  contents. 

To  illustrate,  take  the  following  example. 

464.  Example.      What  is  the  volume  of  a  solid  6  ft.  long,  4  ft. 
high,  and  3  ft.  wide  ? 

SOLUTION.  —  Select  a  measuring  unit.  This  may  be  any  cube,  each  side  of 
which  is  a  unit  of  length.  For  convenience,  take  1  cu.  ft.  (a)  as  a  measuring 
unit.  Then  lines  are  drawn  as  represented  in  (&),  (c)  and  (d).  (&)  contains 
three  times  as  many  cubic  feet  as  (a),  or  3  cu.  ft.  (c)  contains  four  times  as 
many  cubic  feet  as  (?>),  or  12  cu.  ft.,  and  the  entire  figure,  or  (d),  contains  six 


§§  464-469] 


PRACTICAL  MEASUREMENTS 


173 


times  as  many  cubic  feet  as  (c),  or  72  cu.  ft.     Therefore,  the  solid  contents 
of  the  rectangular  solid  given  is  72  cu.  ft.    Hence, 


(d) 


1  cu.  ft. 


TJie  dimensions  of  a  rectangular  solid  reduced  to  units  of  the  same 
denomination  and  multiplied  together  express  the  solid  contents  of  a  solid 
in  cubic  units  of  the  same  denomination  as  the  units  of  length. 

465.  A  cylinder  is  a  circular  body  of  uniform  diameter,  the  bases 
of  which  are  parallel  circles. 

When  all  the  points  of  one  circle  are  equally  distant  from  all  the  points  of 
another  circle,  the  circles  are  said  to  be  parallel  circles. 

466.  The  lateral  surface  of  a  cylinder  is  the  surface  of  its  curved 
sides. 

467.  The  lateral  surface  of  a  cylinder  is  equal  to  the  surface  of  a 
rectangular  body,  the  length  and  height 

of  which  are  equal  to  the  circumference 
and  height  of  the  cylinder. 

Thus,  the  lateral  surface  of  the  cylinder  in 
the  accompanying  diagram  is  the  area  of  the 
rectangle  described  by  A,  B,  C,  and  D  back  of 
the  cylinder.  Hence, 

468.  To  find  the  area  of  the  lateral  surface  of  a  cylinder, 
Multiply  the  circumference  of  the  base  by  the  height  of  the  cylinder. 

469.  To  find  the  solid  contents  of  a  cylinder, 
Multiply  the  area  of  the  base  by  the  height  of  the  cylinder 


174  DENOMINATE   NUMBERS  [§§  469-472 

WRITTEN  EXERCISE 

1.  What  will  be  the  cost  of  a  sheet-iro^  smokestack  40  ft.  high 
and  2  ft.  in  diameter,  at  15  ^  per  square  foot  ? 

2.  A  bin  is  18  ft.  long,  4J  ft.  wide,  and  18  ft.  high.     How  many 
cubic  feet  in  the  bin  ? 

3.  At  15^  per  cubic  yard,,  how  much  will  it  cost  to  excavate  a 
cellar  85  ft.  long,  43  ft.  wide,  and  14  ft.  deep  ? 

4.  How  many  cubic  feet  of  stone  in  a  walk  195  ft.  long,  4£  ft. 
wide,  and  1|  ft.  thick  ? 

5.  How  many  cubic  feet  in  a  cylinder  10  ft.  in  diameter  and 
20ft.  long? 

6.  A  rectangular  bin  contains  259,200  cu.  ft.     If  it  is  40  yd.  long 
and  20  yd.  wide,  how  many  feet  high  is  it  ? 

7.  Find  the  cost  of  digging  a  round  well  25  ft.  deep  and  8  ft.  in 
diameter,  at  35  ^  per  cubic  yard. 


BOARD  MEASURE 

470.  In  measuring  lumber,  boards  one  inch  or  less  in  thickness 
are  estimated  by  the  square  foot. 

Thus,  a  board  18  ft.  long,  12  in.  wide,  and  1  in.  thick  contains  18  sq.  ft.  or 
18  ft.  board  measure. 

471.  In  measuring  lumber  more  than  one  inch  in  thickness  the 
boards  are  estimated  by  the  number  of  square  feet  of  boards,  one 
inch  in  thickness,  to  which  they  are  equal 

Thus,  a  board  12  ft.  long,  12  in.  wide,  and  2J  in.  thick  contains  2$  times 
12  board  ft.,  or  30  board  ft. 

472.  Unless  sawed  to  order,  the  width  of  the  board  is  reckoned 
only  to  the  next  smaller  half-inch,  except  in  che*rry,  black  walnut, 
etc.,  where  the  price  is  15  ^  per  foot  and  upward. 

Thus,  a  board  8J  in.  in  width  is  reckoned  8  in.;  a  board  12|  in.  in  width  is 
reckoned  12£  in. ;  eta 


J§  473-475]  PRACTICAL   MEASUREMENTS  175 

473.  When  the  width  of  a  board  tapers  uniformly,  the  average 
width  is  found  by  finding  one  half  the  sum  of  the  two  ends. 

Thus,  a  tapering  board  16  ft.  long,  12  in.  wide  at  one  end,  and  6  in.  wide  at 
the  other,  and  1  in.  thick,  contains  f  (12 -f  6  -*-  2  -i- 12)  of  16  sq.  ft.,  or  12 
board  ft. 

474.  Examples.    1.    How  many  board  feet  in  6  pieces  of  hemlock, 
2  in.  thick  by  6  in.  wide  by  18  ft.  long  ? 

SOLUTION.      Since  board  feet  are 

8  X  $  X  )3  X  18  =  108  board  ft.      equal  to  square  feet  one  inch  in  thick- 
ness, the  length  of  the  board  in  feet, 

multiplied  by  the  width  of  the  board  in  inches,  and  divided  by  12,  is  equal  to  the 
number  of  board  feet  in  one  board,  one  inch  in  thickness  ;  but  since  the  board 
is  2  in.  in  thickness,  2  times  this  result  is  the  number  of  board  feet  in  each  board, 
and  6  times  this  result,  the  number  of  board  feet  in  the  6  boards. 

To  shorten  the  work,  arrange  the  factors  which  are  to  be  multiplied  together 
as  shown  in  the  margin,  and  mentally  cancel  the  12  in  the  divisor  from  any 
factor  or  factors  in  the  dividend.  2  x  6  in  the  dividend  is  equal  to  the  12  in  the 
divisor;  hence  each  board  contains  18  board  ft.,  and  the  6  boards  108  board  ft. 

&  Find  the  number  of  board  feet  in  6  pieces  of  hemlock  4  in. 
thick  by  5  in.  wide  by  16  ft.  long. 

2  SOLUTION.     Reasoning  as  in  prob- 

$£  lem  1,  6  x  4  x  5  x  16  -*• 12  is  equal  to 

0X^4x5x16  =  160  board  ft.      the  number  of   board   feet  in  the  6 

pieces  of  hemlock.      Observe  that  6 
multiplied  by  4  contains  12  twice.    Then,  2  x  5  x  16  =  160  board  ft.     Hence, 

475.  To  find  the  number  of  board  feet  in  lumber  more  than  one  inch 
thick, 

Express  the  length  in  feet  and  the  width  and  thickness  in  inches. 
The  product  of  these  three  dimensions,  divided  by  12,  is  equal  to  the 
number  of  feet,  board  measure. 

In  charging,  or  billing  lumber,  the  number  of  pieces  are  entered  first,  then 
the  thickness  and  width  in  inches,  then  the  feet  in  length.  For  example,  in 
recording  6  pieces,  4  in.  thick  by  6  in.  wide  and  20  ft.  long,  the  form  would  be 
thus  :  6  pcs,  4  in.  x  6-in.-20  ft.,  and  would  be  called  off  by  the  salesman,  "6 
four-by-sixes-2Q  ft.,"  four-by-sixes  being  the  name  by  which  he  selects  and  sells 
stock. 

Instead  of  writing  "inches"  and  "feet,"  lumber  billing  clerks  use  (")  for 
inches,  and  (')  for  feet;  thus,  3  in.  by  4  in.-l"  ft.  long  is  written,  3"  X  4"-17', 


176  DENOMINATE   NUMBERS  [§475 

ORAL  EXERCISE 

By  inspection,  determine  the  number  of  board  feet  of  lumber  in  : 

1.    5pcs.  3"   x  4"-16f.  11.    lOpcs.  2"x  6"-16'. 

8.    8pcs.  2"   x  6"-20f.  12.     15  pcs.  3"x  8"-16'. 

S.    9  pcs.  4"  x  6"-20'.  13.    25  pcs.  2"  x  6"-20'. 

4.  10  pcs.  5"   x  7"-12f.  14.    50  pcs.  3"  x  4"-18'. 

5.  15  pcs.  S"  xlO"-20'.  Id.  100  pcs.  2"  x  6"-18'. 

6.  10  pcs.  21"  x  8"-18'.  16.      4  pcs.  9"  x  10"-16'. 

7.  20  pcs.  4"   x   6"-16f  17.    24  pcs.  2"  x  4"-20'. 

8.  12  pcs.  8'ff  xlO"-14'.  18.      6  pcs.  4"  x  5"-12'. 

9.  17  pcs.  6"  x  8>f-20f.  19.     10  pcs.  4"  x  6"-16f. 
10.  20  pcs.  3"  x  4"-14'.  m      6  pcs.  2"  x  5"-22'. 

WRITTEN  EXERCISE 

1.  What  is  the  board  measure  of  7  planks,  each  16  ft.  long,  15 
in.  wide,  and  3  in.  thick  ? 

2.  What  will  be  the  cost  of  plank  at  $  18  per  M  that  will  cover 
a  floor  24  ft.  by  13  ft.,  if  the  plank  is  2 J  in.  in  thickness  ? 

S.  What  will  be  the  cost  of  10  sticks  2  in.  by  4  in.,  10  sticks 
2  in.  by  6  in.,  10  sticks  4  in.  by  4  in.,  and  10  sticks  2  in.  by  10  in.,  if 
they  are  each  16  ft.  long  and  the  cost  is  $15  per  M  ? 

4.  What  will  be  the  cost  at  $  15  per  M  of  a  tapering  board  18  ft. 
long,  1  in.  thick,  and  1\  in.  wide  at  one  end  and  16£  in.  wide  at  the 
other  ? 

5.  Find  the  amount  of  the  following  bill  of  hemlock,  price  by 
the  M  ft.  board  measure : 

26  pcs.  2"  x  6"-18'  at  $  12 ;         24  pcs.  3"  x4"-20'  at  $  15 ; 
128  pcs.  8"  x  4"-14'  at  $20. 

NOTE.     Always  try  to  shorten  the  work  by  mentally  eliminating  the  12's 
from  the  dividend. 

6.  Find  the  amount  of  the  following  bill  of  lumber,  price  by  the 
thousand  ft.  board  measure : 

18  pcs.    3"  x  4"-20'  at  $14;     10  pcs.  2£rr  x  6"-18f  at  $15; 
25  pcs.    4"  x  6"-16'  at  $12;     12  pcs.    3"  x  5"-20'  at  $18. 


§§475-478]  PRACTICAL   MEASUREMENTS  177 

7.   At  $ 32.50  per  M,  what  will  be  the  cost  of: 
8  scantlings  3"  x  4"-18' ;  12  scantlings  4"  x  5"-16' ; 

8  scantlings  5"  x  6"-14f. 

8*   At  $  19.50  per  M,  what  will  be  the  total  cost  of : 
9  boards    1"  x    2"-14' ;  6  boards  1£"  X  18"-16f ; 

15  boards    2"  x  14"-20';  8  boards  1J"  x  12"-18'. 

9.  At  $  24  per  M,  what  will  be  the  cost  of  the  lumber  required 
to  inclose  a  field  40  rd.  square  with  a  board  fence  if  the  boards  are 
15  ft.  long,  5  in.  wide,  and  1  in.  thick,  and  the  fence  5  boards  high  ? 
10.  At  $21.50  per  M,  what  will  be  the  cost  of  the  lumber  in  a 
line  fence  160  rd.  long  if  the  boards  are  11  ft.  long,  7  in.  wide,  and 
1  in.  thick,  and  the  fence  4  boards  high  ? 

WOOD  MEASURE 

476.  A  pile  of  wood  8  ft.  long,  4  ft.  wide,  and  4  ft.  high  is  called 
a  cord. 

477.  A  pile  of  wood  1  ft.  long,  4  ft.  wide,  and  4  ft.  high,  or  |  of 
a  cord,  is  called  a  cord  foot 


478.    Example.     Find  the  number  of  cords  of  wood  in  a  pile  25  ft. 
long,  4  ft.  wide,  and  6  ft.  high. 

3  SOLUTION.    The  product  of  the  dimen- 

25  X  4  X  0      75  sions  of  the  pile  is  equal  to  the  number 

-  =  4      cords- 


1  of  cubic  feet  of  wood.     Since  a  cord  of 

wood  is  8  ft.  long,  4  ft.  wide,  and  4  ft. 
high,  or  contains  128  cu.  ft.,  the  number 
of  cords  in  the  pile  is  found  by  dividing 
this  product  by  128.     Therefore,  the  required  result  is  4^  cords.     Hence, 


178  DENOMINATE   NUMBERS  [§§479-480 

479.  To  find  the  number  of  cords  of  wood  in  any  pile, 

Divide  the  product  of  the  length,  width,  and  height,  expressed  in 
cubic  feet,  by  128. 

WRITTEN  EXERCISE 

1.  How  many  cords  of  wood  in  a  pile  108  ft.  long,  7  ft.  9  in.  high, 
and  6  ft.  wide  ? 

2.  From  a  pile  of  wood  71  ft.  6  in.  long,  9  ft.  4  in.  wide,  and  6  ft. 
8  in.  high,  21f  cords  were  sold.     What  was  the  length  of  the  pile 
remaining  ? 

3.  At  $4.75  per  cord,  what  will  it  cost  to  fill  with  wood  a  shed 
34  ft.  long,  18  ft.  wide,  and  10  ft.  high  ? 

4.  A  pile  of  wood  built  10  ft.  high  and  25  ft.  wide  must  be  how 
long  to  contain  125  cords  ? 

5.  A  pile  of  wood  30  ft.  long,  4  ft.  wide,  and  8  ft.  high  was  sold 
at  $  6  per  cord.     How  much  was  received  for  it  ? 

PAVING 

480.  The  unit  of  paving  is  the  square  foot  or  the  square  yard. 

WRITTEN   EXERCISE 

1.  Find  the  cost  of  paving  a  court  150  ft.  long  and  120  ft.  wide 
at  $  3  per  square  yard. 

2.  Which  would  be  the  more  economical  way  to  pave  a  street 
3  mi.  long  and  1  rd.  wide:  with  granite  blocks  at  $3.65  per  square 
yard,  or  with  asphalt  costing  23^  per  square  foot,  and  how  much 
would  be  saved  ? 

3.  Which  would  be  the  more  economical,  and  how  much :  to  pave 
a  walk  with  stone  at  22^  per  square  foot,  or  with  brick  at  $  1.02  per 
square  yard,  if  the  width  of  the  walk  is  4  ft.  and  the  length  200  ft.  ? 

4.  How  many  granite  blocks  12  in.  by  18  in.  will  be  required  to 
pave  a  mile  of  roadway  42  ft.  in  width  ? 

5.  A  street  4975  ft.  long  and  40  ft.  wide  was  paved  with  Trinidad 
asphaltum  at  $2.G5  per  square  yard.     What  was  the  cost  ? 


§§481-491]  PRACTICAL   MEASUREMENTS  179 

GAUGING 

481.  The  process  of  finding  the  contents  of  any  regular  vessel  in 
gallons,  barrels,  bushels,  etc.,  is  called  gauging. 

482.  In  every  liquid  gallon  there  are  231  cubic  inches.     Hence, 

483.  To  find  the  exact  number  of  gallons  in  a  vessel, 
Divide  the  number  of  cubic  inches  in  the  vessel  by  231. 

484.  In  every  bushel,  stricken  measure,  there  are  2150.42  cubic 
inches.     Hence, 

485.  To  find  the  exact  number  of  stricken  bushels  in  any  bin, 
Divide  the  number  of  cubic  inches  in  the  bin  by  2150.42. 

486.  In  every  bushel,  heaped  measure,  there  are  2747.71  cubic 
inches.     Hence, 

487.  To  find  the  exact  number  of  heaped  bushels  in  any  bin, 

Divide  the  number  of  cubic  inches  in  the  bin  by  2747.71. 

488.  Approximate  Rules.     For  practical  purposes  the  rules  given 
below  generally  approximate  accuracy. 

489.  Example.    What  decimal  of  a  stricken  bushel  is  1  cu.  ft.  ? 

O    I 

_  SOLUTION.     Since    in    every    bushel    there    are 


2150.42)1728.000      2150.42  cu.  in.,  and  in  every  cubic  foot  1728  cu.  in., 
1720336      a  cubic  foot  k  !Bo£$»  or  approximately  .8  of  a 
^bushel.    Hence, 


490.  To  find  the  approximate  number  of  stricken  bushels  in  any 
number  of  cubic  feet, 

Multiply  by  .8  ;  and, 

To  find  the  approximate  number  of  cubic  feet  in  any  number  of 
stricken  bushels, 
Divide  by  .8. 

491.  Example.   What  decimal  of  a  cubic  foot  is  a  heaped  bushel  ? 
2747.71)1728.0000(.63—  SOLUTION.     Since  a  cubic  foot  contains 

1648  626  1728  cu*  in>'  an(*  a  neaPed  bushel  2747.71  cu.  in., 

70  3740  a  cubic  foot  may  be  reduced  to  a  decimal  of  a 

82  4313  heaped  bushel  as  shown  in  the  margin.     The 

result  is  .63-  cu.  ft.     Hence, 


180  DENOMINATE   NUMBERS  [§§  492-494 

492.  To  find  the  approximate  number  of  heaped  bushels  in  any 
number  of  cubic  feet, 

Multiply  by  .63;  and, 

To  find  the  approximate  number  of   cubic  feet  in  any  number  oi 
heaped  bushels, 

Divide  by  .63. 

493.  Example.   How  many  gallons  in  a  cubic  foot  ? 

1728  -T-  231  =  7.48+  SOLUTION.     Since  a  gallon  contains  231  cu.  in., 

and  a  cubic  foot  1728  cu.  in.,  the  number  of  gallons 
in  a  cubic  foot  is  found  by  dividing  1728  by  231.     The  result  is  7.48+.     Hence, 

494.  To  find  the  approximate  number  of  gallons  in  a  cistern, 

Multiply  the  number  of  cubic  feet  by  7~2)  and  from  the  product  sub- 
tract -^  of  the  product. 

ORAL   EXERCISE 

1.  Find  the  approximate  number  of  bushels  of  grain  contained 
in  a  box  that  is  5  ft.  long,  4  ft.  wide,  and  3  ft.  high. 

2.  How  high  must  a  box  10  ft.  long  and  6  ft.  wide  be  built  to 
hold,  approximately,  240  bu.  ? 

3.  A  vat  11  in.  long,  7  in.  high,  and  3  in.  wide  will  contain  how 
many  gallons  of  water  ? 

4.  A  vat  containing  2  gal.  of  water  is  14  in.  long  and  11  in.  wide. 
How  high  is  it  ? 

WRITTEN  EXAMPLES 
Find  the  approximate  number  of  bushels  of  grain  required  to  fill : 

1.  A  bin  18  ft.  x  6  ft.  x  4  ft.  4.   A  bin  25  ft.  x  12  ft.  x  8  ft. 

2.  A  bin  13  ft.  x  20  ft.  x  5  ft.         5.   A  bin  12J  ft.  x  8  ft.  x  6  ft. 

3.  A  bin  20  ft.  x  8  ft.  x  6  ft.  6.   A  bin.20  ft.  x  12  ft.  x  8  ft. 

7.   A  cubical  cistern  is  10  ft.  on  a  side.     Find  the  exact  number 
of  barrels  of  31^  gal.  each  that  it  will  contain. 


§§  494-000]  PRACTICAL  MEASUREMENTS  181 

8.  A  farmer  exactly  filled  a  bin  9  ft.  wide,  12  ft.  long,  and  7|  ft. 
deep,  with  wheat  grown  from  a  field  yielding  32J  bushels  per  acre. 
How  long  was  the  field  if  its  width  was  50  rd.  ? 

9.  Find  the  exact  number  of  gallons  of  water  in  a  well  6  ft.  in 
diameter,  when  the  water  is  9  ft.  in  depth. 

10.  A  cask  is  24  in.  at  the  chime,  30  in.  at  the  bung,  and  3  ft.  long. 
Find  the  exact  number  of  gallons  that  may  be  put  into  it,  if  the  cask 
is  already  f  full. 

STONE  AND  BRICK  WORK 

495.  The  unit  of  stone  work  is  the  cubic  yard  or  the  perch. 

496.  A  perch  of  stone  or  masonry  is  a  rectangular  solid  16J  ft. 
long,  1|  ft.  wide,  and  1  ft.  high,  and  contains  24J  cu.  ft. 

The  number  of  cubic  feet  allowed  for  a  perch  of  stone  or  masonry  varies  in 
different  localities.  In  some  places  it  is  considered  16£  cu.  ft.,  and  in  other, 
places,  25  cu.  ft. 

497.  The  number  of  bricks  in  walls  is  usually  estimated  by  the 
thousand,  and  22  common  bricks  laid  in  mortar  are  counted  for  each 
cubic  foot  of  wall. 

A  common  brick  is  8  in.  long  by  4  in.  wide  by  2  to.  thick, 

498.  In  making  estimates  for  stone  and  brick  work,  masons  take 
girt  measurements.     Whether  anything  is  to  be  deducted  for  the 
area  of  the  windows  and  other  openings  is  generally  fixed  by  con- 
tract.    In  some  localities  one  half  the  area  of  such  openings  is 
always  deducted,  while  in  others  nothing  whatever  is   deducted. 
In  estimating  material,  however,  allowance  is  generally  made  for 
the  corners  and  all  openings. 

In  taking  girt  measurements  the  corners  are  counted  twice,  but  this  is  con- 
sidered offset  by  the  extra  work  required  in  building  corners ;  the  work  around 
openings  is  also  more  difficult  than  straight  work. 

499.  To  find  the  number  of  perches  of  stone  or  masonry  in  a  wall, 
Divide  the  contents  of  the  wall  in  cubic  feet  by  2$. 

500.  To  find  the  number  of  bricks  for  a  wall, 
Multiply  the  number  of  cubic  feet  in  the  wall  by  2$. 


132  DENOMINATE   NUMBERS  [§  600 

WRITTEN  EXERCISE 

1.  A  cellar  is  24  ft.  square  inside  of  the  wall,  which  is  9  ft.  high 
and  2  ft.  thick.     How  many  perches  of  24|  cu.  ft.  each  would  a 
mason  estimate  for  the  wall? 

2.  How  many  cubic  yards  of  masonry  in  the  foundation  walls  of 
a  house  50  ft.  long  and  30  ft.  wide,  outside  measurements,  if  the  wall 
is  uniformly  2  J  ft.  wide  and  8  ft.  high  ? 

S.  How  many  perches  of  stone,  actual  measure,  will  be  required 
to  inclose  a  field  32  rd.  long  and  24  rd.  wide,  with  a  wall  4J  ft.  high 
and  2^-  ft.  thick,  counting  25  cu.  ft.  to  the  perch  ? 

4.  What  will  be  the  cost,  by  mason's  measure,  of  building  the 
walls  of  a  block  140  ft.  long,  66  ft.  wide,  and  47  ft.  high,  outside 
measurements,  at  $1.45  per  perch  of  24|  cu.  ft.,  if  the  walls  are 
18  in.  thick-  and  no  allowance  is  made  for  openings  ? 

5.  How  many  common  bricks  will  be  required  to  erect  the  walls 
of  a  flat-roofed  building  120  ft.  long,  85  ft.  wide,  and  22  ft.  high,  out- 
side measurements,  if  the  walls  are  18  in.  in  thickness  and  an  allow- 
ance of  600  cu.  ft.  is  made  for  openings  ?     (Solve  (1)  by  mason's  meas- 
ure, making  allowance  for  the  openings,  and  (2)  by  actual  measure.) 

6.  At  $2.25  per  perch  of  24|  ft.,  how  much  will  it  cost,  by 
mason's  measure,  to  build  the  walls  for  a  building,  the  length  of 
which  is  49.5  ft.  and  the  width  24|  ft.,  the  walls  to  be  14  ft.  high 
from  the  foundation  and  18  in.  thick  ? 

7.  How  many  common  bricks  will  be  required  in  building  the 
four  walls  for  a  building  90  ft.  long,  50  ft.  wide,  and  60  ft.  high, 
outside  measurements,  if  the  walls  are  uniformly  1J  ft.  thick,  and 
340  cu.  ft.  is  allowed  for  openings  ?     (Solve  (1)  by  mason's  measure, 
making  allowance  for  the  openings,  and  (2)  by  actual  measure.) 

WRITTEN  REVIEW 

/.  In  estimating  the  number  of  posts  necessary  for  a  wire  fence 
to  inclose  a  rectangular  field  120  rd.  long,  it  is  found  that  to  put  the 
posts  12  ft.  instead  of  16J  ft.  apart  will  require  180  more  posts. 
What  is  the  field  worth  at  $90  per  acre  ? 

2.  How  high  must  wood  be  piled  in  a  shed  which  is  28   ft.  long 
and  16  ft.  wide,  to  contain  28  cords  ? 

3.  What  is  the  cost,  at  $  90  per  acre,  of  a  rectangular  farm  having 
a  length  twice  its  width,  if  the  perimeter  is  480  rd.  ? 


§500]  PRACTICAL  MEASUREMENTS  183 

4.  How  much  would  it  cost  to  plaster  the  walls  and  ceiling  of  a 
room  25  ft.  long,  18  ft.  wide,  and  12  ft.  high,  at  27^  per  square  yard, 
making  an  allowance  of  396  ft.  for  doors,  windows,  etc.  ? 

5.  What  will  be  the  cost,  at  $  12  per  M,  of  the  boards  required  for 
a  sidewalk  32  rd.  long  and  4  ft.  wide,  if  the  boards  are  16  ft.  long. 
1  in.  thick,  and  8  in.  wide  ? 

6.  Express  £  15.6  in  dollars  and  cents. 

7.  At  1|^  per  square  inch,  what  will  it  cost  to  bronze  a  cube  the 
depth  of  which  is  2  ft.  ? 

8.  I  bought  a  farm  200  rd.  long  for  $3600.     If  I  paid  $  72  an 
acre  for  the  farm,  how  much  will  it  cost  to  fence  it  at  25^  per  rod  ? 

9.  What  will  be  the  cost  of  excavating  a  cellar  45  ft.  long,  30  ft. 
wide,  and  8  ft.  deep,  at  35^  per  cubic  yard  ? 

10.  Estimating  that  150  cu.  ft.  of  air  should  be  allowed  for  each 
pupil,  how  many  pupils  can  be  accommodated  in  a  schoolroom  45  ft 
long,  30  ft.  wide,  and  10  ft.  high  ? 

11.  At  9^  per  square  yard,  what  will  it  cost  to  paint  the  four 
sides  and  bottom  of  a  tank  10  yd.  long,  16  ft.  wide,  and  18  ft.  deep. 

12.  At  $  125  per  acre,  find  the  difference  in  cost  between  two 
fields,  the  first  of  which  contains  80  sq.  rd.,  and  the  second  of  which 
is  80  rd.  square. 

18.  Bought  of  a  produce  dealer  900  pounds  of  wheat  at  $  1  per 
bushel,  and  gave  in  payment  a  pile  of  wood  16  ft.  long,  6  ft.  high, 
and  4  ft.  wide.  What  price  per  cord  did  I  receive  for  the  wood  ? 

14.  How  many  cubic  feet  in  8  pieces  of  hemlock  24  ft.  long, 
14  in.  wide,  and  8  in.  thick  ?     How  many  board  feet  ?     What  part 
of  a  cubic  foot  is  a  board  foot  ? 

15.  A  man  bought  a  piece  of  land  80  rd.  square,  and  after  retail- 
ing 240  sq.  rd.,  sold  the  remainder  at  $90  per  acre.     How  much  did 
he  receive  ? 

16.  Which  would  be  the  more  economical,  and  how  much:  to  pave 
a  sidewalk  1  mi.  long  and  1  rd.  wide  with  asphalt  costing  21  ^  per 
square  foot,  or  with  granite  blocks  costing  $  2.95  per  square  yard  ? 

17.  What  will  it  cost  at  $2.50  per  yard  to  carpet  a  floor  24  ft. 
long  by  17  ft.  wide,  if  the  strips,  which  are  {  of  a  yard  wide,  are  run 
lengthwise  of  the  room,  and  there  is  a  waste  of  9  in.  on  each  strip  for 
matching  the  pattern  ? 


PERCENTAGE  AND  ITS  APPLICATIONS 

PERCENTAGE 

501.  The  arithmetical  processes  in  which  the  basis  of  comparison 
is  one  hundred  are  termed  percentage. 

502.  Per  cent,  usually  written  "  %,"  is  an  abbreviation  of  the 
Latin  words  "  per  centum,"  and  signifies  by  the  hundred. 

Thus,  eight  per  cent  means  eight  of  every  one  hundred  parts,  or  .08,  and  is 
written  8  % ;  seven  and  one  half  per  cent  means  seven  and  one  half  of  every 
one  hundred  parts,  or  .07|,  and  is  written  7|  %. 

503.  The  essential  elements  of  percentage  are  the  base,  the  rate, 
and  the  percentage. 

504.  The  base  is  the  number  upon  which  the  percentage  is  com- 
puted. 

505.  The  rate  is  the  number  of  hundredths  of  the  base  to  be 
taken ;  it  is  usually  expressed  as  a  decimal. 

506.  The  percentage  is  the  result  obtained  by  taking  a  certain 
per  cent  of  the  base ;  or, 

It  is  the  product  obtained  by  multiplying  the  base  by  the  rate. 
In  the  expression  " 5% of  500  is  25,"  the  base  is  600 ;  the  rate,  6% ;  and  the 
percentage,  25. 

507.  The  amount  per  cent  is  100%  increased  by  the  rate ;  or,  1 
plus  the  rate,  expressed  as  a  decimal. 

508.  The  difference  per  cent  is  100%  diminished  by  the  rate ;  or, 
1  minus  the  rate  expressed  as  a  decimal. 

509.  The  amount  is  the  base  plus  the  percentage. 

510.  The  difference  is  the  base  minus  the  percentage. 

511.  General  Principles.     The  base  may  either  be  an  abstract  or 
a  denominate  number;  the  rate  per  cent  must  always  be  an  abstract 
number;   and  the  percentage,  amount,  and  difference  always  huvo 
the  same  name  as  the  base. 

184 


§512] 


PERCENTAGE 


185 


512.  Since  a  per  cent  is  a  number  of  hundredths,  it  may  be 
expressed  either  as  a  decimal  or  as  a  common  fraction.  The  prin- 
ciples of  aliquot  parts  may  therefore  be  used  to  advantage  in  many 
operations  in  percentage  and  its  applications. 

TABLE 


PER  CENT 

DECIMAL 
VALUE 

FRACTIONAL 
VALUE 

PART  OF 

100%, 

OR  THE  BASE 

PER  CENT 

1  >EOIMAL 

VALUE 

FRACTIONAL 
VALUE 

PART  OF 
100%, 
OR  THE  BASE 

1% 

.01 

1 
100 

Ih 

221  % 

.22| 

22J 

100 

| 

H% 

.01| 

11 

100 

'So 

28$% 

.28? 

28{ 

100 

i 

11% 

ou 

ii 

i 

3U°/ 

31  1 

§11 

A 

s% 

1 

100 

** 

?  /o 

I 

100 

16 

21% 

.021 

Jl 

100 

A 

331% 

.331 

100 

1 

31% 

.031 

Ji 

100 

* 

371% 

.871 

100 

1 

6}% 

.06^ 

100 

A 

42?% 

.42? 

100 

? 

6|% 

•06f 

100 

A 

48}% 

.43f 

43| 
100 

A 

81% 

.081 

JL 
100 

A 

60% 

.60 

ao 

100 

1 

%% 

.09^ 

100 

A 

564% 

.56} 

100 

A 

10% 

.10 

10 

100 

A 

621% 

.621 

100 

! 

n*% 

.111 

111 

100 

i 

66|% 

•66f 

66| 
100 

1 

121% 

.121 

•'a 

100 

i 

681% 

.68} 

68| 
100 

H 

14?% 

.14? 

100 

» 

75% 

.75 

76 
100 

1 

16f% 

.16| 

Too 

i 

8H% 

.81} 

100 

if 

18}% 

.18| 

100 

A 

831% 

.831 

100 

i 

20% 

.20 

20 
100 

r 

871% 

.871 

871 
100 

i 

25% 

.25 

26 

100 

3 

83}% 

.931 

93| 
100 

H: 

186                     PERCENTAGE   AND   ITS  APPLICATIONS  {_§§  512-516 

DRILL  EXERCISE 

What  common  fraction  in  its  simplest  form  is  equivalent  to  : 

1.  1%  ?          5.  20%  ?           9,  100%  ?      18.  16f  %  ?  17.  125%  ? 

8.  4%  ?          ft  25%  ?         m  61%  ?        ^.  331%  ?  U.  150%  ? 

3.  5%  ?          7.  50%  ?         ^.  81%  ?        ^5.  66|%  ?  m  21%  ? 

£  10%  ?       A  75%  ?         12.  121%  ?      ig.  871%  ?  m  If  %  ? 

Express  decimally  : 

21.  28%.                 ££.  101%.               27.  6|%.  50.  182%. 

22.  35%.                 05.  7f%.                 £$>.  250%.  81.  415%. 
£9.  50%.                 «ft  9|%.                 m  137%.  ^.  106%. 

Express  as  a  rate  per  cent  : 

88.                86.       .          57.     .            *P.                 -y.  ^.       - 


513.  The  operations  of  percentage  are  based  upon   the  same 
general  principles  as  the  operations  of  simple  multiplication  and 
division,  the  base  in  percentage  corresponding  to  the  multiplicand 
in  simple  multiplication,  the  rate  to  the  multiplier,  and  the  percentage 
to  the  product:    Hence,  any  two  of  the  elements  of  percentage  being 
given,  the  other  may  be  found. 

514.  The  formulse  for  percentage  are  derived  from  the  funda- 
mental principles  of  multiplication  and  division,  as  follows  : 

1.  Multiplicand  x  multiplier  =  product  ;    hence,   base  x  rate  per 
cent  =  percentage. 

2.  Product  -T-  multiplicand  =  multiplier;    hence,  percentage  -+-  base 
=  rate  per  cent. 

3.  Product  •*-  multiplier  =  multiplicand;    hence,  percentage  -r-  rate 
per  cent  =  base. 

515.  To  find  the  percentage,  the  base  and  rate  being  given. 

516.  Examples.     1.   What  is  9%  of  $500? 

$500  base. 

nQ  SOLUTION.    9%  of  a  number  is  .09  of  it.    There- 

fore, 9%  of  $500  is  .09  of  $500,  or  $45. 


$45.00  percentage. 


§§  516-617]  PERCENTAGE  18? 

2.    What  is  121%  of  $888.80?  SOLUTIONS.     («)  12|%  of  a  num- 

ber is  .126  of  it;  therefore,  12i% 

(a)  $888.80  x  .125%=  $111.10.      Of  $888.80  is  .125  of  $888.80,  or 

$111.10.     Or, 

(6)  .125%  of  a  number  is  .12$,  or 

(6)  \  Of  $888.80  =  $111.10.  i  of  it;  therefore,  \  of  $888.80, 

or  $111.10,  is  the  required  result. 

517.   Hence,  to  find  a  percentage  of  a  number, 

Multiply  the  base  by  the  given  rate  per  cent,  expressed  decimally.  Or, 

Take  such  apart  of  the  base  as  the  rate  per  cent  is  of  100%. 

DRILL  EXERCISE 

1,  Formulate  a  short  method  for  finding  8  \  %  of  a  number. 
SOLUTION.     Since  8£%  of  a  number  is  .08|  or  ^  of  it,  to  find  8J%  of  a  num- 

ber, divide  by  12. 

2.  Give  a  short  method  for  finding  the  percentage  when  the  base 
is  given  and  the  rate  is  121%;  16f%;  25%;  331%;  9Ti-%;  28$%. 

S.   Formulate  a  short  method  for  finding  75%   of  a  number? 
18f  %  of  it;  43f  %  of  it;  62|%  of  it;  31^%  of  it. 

4.  What  aliquot  part  of  a  number  is  93f  %  of  it?  83£%  cf  it? 
66f  %  of  it  ?    22f  %  of  it  ?    871%  of  it  ? 

5.  Express  as  a  rate  per  cent  :   -fa  of  a  number  ;   £  of  a  number  ; 
|  of  a  number  ;  -|  of  a  number  ;  -^  of  a  number  ;  -^  of  a  number  ; 
^j  of  a  number;  -fa  of  a  number;  f  of  a  number:  ^  of  a  number. 

6.  Give  a  short  method  for  finding  1^%  of  a  number. 
SOLUTION.     1|%  of  a  number  is  .01$,  or  -fa  of  it.    Hence,  to  find  1|%  of  a 

number,  point  off  one  place  and  divide  by  8. 

7.  Give  a  short  method  for  finding  1|%  of  a  number;   2|%; 

;   561%;   6|%;   20%. 


ORAL  EXERCISE 

By  inspection,  find  the  value  of  : 

1.  37i%  of  160.             7.  66|%of930.  13.  18|%of480. 

8.  62|%of320.            8.  2|%  of  360.  14.  31  J%  of  320. 

8.  8|%  of  720.  .            9.  331%  of  930.  15.  43f  %  of  160. 

4.  6i%of960.             10.   121%  of  880.  16.  56j%of800. 

5.  142%  of  210.           11.   6|%  of  450.  17.  621%  of  240. 

6.  25%  of  680.             12.   16f  %  of  666.  18.  75%  of  128. 


188  PERCENTAGE   AND   ITS  APPLICATIONS  [§61b 

SHORT  METHODS 

518.  Examples.    1.  What  is  36%  of  $2500? 

if  <n.  ogQQ  __  <jfc  9QQ  SOLUTION.    Since  36  times  25  win  give  the 

same  product  as  25  times  36,  36%  of  $2500  will 

give  the  same  result  as  25%  of  $3600:    25%  is  |  of  a  number  j  therefore,  \  of 
$3600,  or  $900,  is  the  required  result. 

2.  What  is  16%  of  $12,500? 

4  of  16  000  =  $  2000          SOLUTION.    16  times  12f  will  give  the  same  product 
as  12J  times  16 ;  hence,  16%  of  $  12,500  is  equivalent 

to  12|%  of  $16,000.    12|%  is^  of  a  number;  £  of  $16,000  is  $2000,  or  the 
required  result. 

8.  What  is  24%  of  $37,500? 

24  000  X  &  =  $  9000  SOLUTION.    24  times  37|  will  give  the  same  product 

as  37£  times  24  ;  hence,  24%  of  $37,500  is  equivalent 

to  37|%  of  $24,000.    37  J%  is  f  of  a  number;  f  of  $24,000  is  $9000,  or  the 
required  result. 

ORAL  EXERCISE 

By  inspection,  find  the  value  of; 

1.  12%  of  $2500.  7.  24%  of  $750.  13.  24%  of  $62,500 

2.  16%  of  $2500.            8.   12|%  of  $960.  /£  32%  of  $3750. 
S.  44%  of  $7500.            9.  56%  of  $1250.  15.  192%  of  $875. 

4.  48%  of  $1250.          10.  16%  of  $3125.        16.   125%  of  $888. 

5.  250%  of  $64.  11.  32%  of  $5625.        17.   1250%  of  $640 
ft  75%  of  $160.  12.  16|%of$360.        18.  96%  of  $250. 

WRITTEN  EXERCISE 

1.  Having  $  240,000  to  invest,  a  gentleman  bought  United  States 
bonds  with  33^  %  of  it,  a  home  with  25  %  of  the  remainder,  and 
invested  what  still  remained  equally  in  farming  lands  and  manufac- 
turing stock.     How  much  did  he  invest  in  manufacturing  stock  ? 

2.  A  collector  deposited  $  2400  in  coin  and  12J  %   as  much  in 
bank  bills.     What  was  the  amount  of  his  deposit  ? 

8.  In  a  certain  barn  there  are  24,960  bu.  of  grain,  of  which  33J  % 
Is  wheat,  12J  %  oats,  and  the  remainder,  barley.  How  many  bushels 
of  each  kind  of  grain  are  there  in  the  barn  ? 

4" 


§§  518-521]  PERCENTAGE  189 

4.  A  jobber  having  2160  bags  of  coffee,  sold  at  one  time  8J  %,  at 
another,  25  %  of  what  remained,  and  at  a  third,  sold  33J  %  of  what 
still  remained.     Find  the  value  of  what  was  still  left  at  $  18  per  bag. 

5.  A  farmer  having  156  sheep  to  shear,  agreed  to  pay  for  their  ^N 
shearing  4%  of  the  sum  received  for  their  wool.     If  the  fleeces  aver- 
age 1\  Ib.  each,  and  are  sold  at  30^  per  pound,  how  much  was  paid 
for  shearing  ? 

6.  A  dealer  having  bought  240  doz.  eggs  at  25^  per  dozen,  sold 
8J  %  of  them  at  cost  and  the  remainder  at  27^  per  dozen.     What 
was  his  profit  ?  ; ,  • 

7.  A  farmer  having  raised  1240  *bu.  wheat,  used  5%  of  it  for 
seed  and  5  %  of  it  for  bread.     He  then  sold  to  one  man  33 \  %  of  the 
remainder  at  $  1  per  bushel,  and  to  another  25  %  of  what  still  remained 
at  $1.10  per  bushel.     How  much  was  received  from  both  sales,  and 
how  many  bushels  were  left  unsold  ? 

8.  A  man  owning  an  estate  of  $  200,000  bequeathed  10  %  of  it 
to  a  college,  10  %  of  the  remainder  to  a  church,  and  divided  what 
still  remained  equally  among  his  four  children.    What  did  each 
child  receive? 

519.  To  find  the  rate,  the  base  and  percentage  being  given. 

520.  Example.    A  farmer  having  480  bu.  of  wheat,  sold  120  bu. 
What  per  cent  of  his  wheat  did  he  sell  ? 

(a) 

SOLUTION,     (a)  Since  the  percentage  is  the  product 

*^  "*"  ^°^  =st  *****      of  the  base  and  rate,  the  quotient  obtained  by  dividing 

/r  the  percentage  by  the  base  will  be  the  rate.    Or, 

(0) 

120  _  i  (&)  12°  is  Iffc  or  J»  of  480>  and  since  48°  is  10° % 

of  itself,  120,  which  is  1  of  480,  must  be  1  of  100%, 
J  of  100%  =25%.     Or250/o.    or, 

/c\  (c)  Since  480  is  100  %  of  itself,  1  %  of  480  would  be 

T&Q  part  of  it,  or  4.80.     Since  4.80  is  1  %of  480,  120 

=  ***•       would  be  as  many  times  1  %  as  4.80  is  contained  times 

25  times  1  %  =  25  %.  in  120'  which  is  25  times;  26  times  1%  is  25%* 

521.  Hence,  to  find  the  rate, 

Divide  the  percentage  by  the  base  and  express  the  number  of  hun- 
dredths  obtained  as  a  rate  per  cent. 


190  PERCENTAGE  AND  ITS  APPLICATIONS  [§521 

f 

ORAL  EXERCISE 
What  per  cent  of : 

1.  $50  are  $5?  5.  12  da.  are  4  da.?  9.  .12  are  .24? 

&  15  hr.  are  45  hr.?  ft  -J-  bu.  is  J  bu.  ?  10.  .24  are  .12  ? 

3.  24  bu.  are  48  bu.?  7.  £  yd.  is  -^  yd.?  ^7.  48  hr.  are  36  hr.? 

4.  15  A.  are  5  A.?  8.  2.4  are  3.6  ?  jflR  $  160  are  $  40  ? 

WRITTEN  EXERCISE 

1.  Of  a  stock  of  800  yd.  of  prints  240  ygl.  were  sold  at  one  time 
and  160  yd.  at  another.     What  per  cent  of  the  whole  stock  was  still 
unsold  ? 

2.  Of  a  regiment  of  men  entering  battle  1040  strong  only  260     ^ 
came  out  unhurt,  \  of  the  remainder  having  been  killed.     What  per    ' 
cent  of  the  whole  regiment  was  killed  ? 

S.  Out  of  900  bu.  of  potatoes  put  in  storage  October  15,  45  bu. 
were  found  unsound  April  1.  What  per  cent  of  the  whole  was 
sound  ? 

4.  A  merchant's  profits  for  1903  were  $3800,  or  $200  in  advance  ^ 
of  his  profits  for  1902.     Find  the  per  cent  of  increase  in  the  profits 
of  1903  over  those  of  1902. 

5.  From  a  cask  of  lard  containing  320  Ib.  70  Ib.  were  sold  at  one  ~ 
time  and  30%  of  the  remainder  at  another.     What  per  cent  of  the 
whole  remained  unsold  ? 

ft  In  a  certain  school  there  are  1800  male  pupils  and  200  female 
pupils.  What  per  cent  more  are  the  male  than  the  female  pupils  ? 

7.  A  merchant  failed  in  business,  having  resources  amounting* 
to  $  15,000  and  liabilities  amounting  to  $  75,000.     What  per  cent  oi 
his  debts  can  he  pay  ? 

8.  What  per  cent  more  is  £  than  \  ? 

9.  What  per  cent  less  is  |  than  \  ? 

10.    A  has  25%  less  money  than  B.     What  per  cent  has  B  more 
than  A  Y 


§§  522-624]  PERCENTAGE  19l 

522.  To  find  the  base,  the  percentage  and  rate  being  given. 

523.  Example.    375  is  12%%  of  what  number  ? 

(a)  SOLUTIONS,     (a)  12$%  of  a  number  Is 

3  00Q  equal  to  .125  of  it.    If  .125  of  a  number  is 

-OPN  Q7f-  rvnrv  876,  the  whole  number  may  be  found  by 

the  principles  of  division  of  decimals  (305; 
Or 

(&)  (6)  If  12$%  of  a  number  is  376,  1%  of 

12£%  =  375.  the  number  is  &  of  375,  or  30,  and  100%  or 

1  %  ==  30.  the  whole  of  the  number  is  100  times  30,  or 

100$  =  3000.  800°-    Or' 

(c)  (c)  12$%  of  a  number  is  $  of  it.    If  $ 

4  01  ...  ,          orrw    of  a  number  is  375,  4,  or  the  whole  number. 

12|%,  or  I,  of  a  number =375.  .g  8  timeg  ^  ^^ 

f  =  8  times  375,  or  3000.        required  result  is  3000. 

524.  Hence,  to  find  the  base, 
Divide  the  percentage  by  the  rate. 


ORAL 

?.  18  is  £  of  what  number  ?  6.  Of  what  number  is  12£  5%  ? 

&  Of  what  number  is  16  25%  ?  &  555  is  5%  of  what  number  ? 

S.  3215  is  41^%  of  what  number  ?  7.  19  is  16f  %  of  what  number? 

4.  Of  what  number  is  125  62£%  ?  8.  90  is  £  of  what  number  ? 

P.   A  man's  yearly  expenses  are  $150,  or  12^%  of  his  income. 
What  is  his  income  ? 

WRITTEN  EXERCISE 


1.  2i%  of  240  is  25%  of  what  number? 

2.  On  a  bill  of  $1280  $64  discount  was  allowed.    What  was        ^  J 
the  per  cent  of  discount  ? 

3.  Jan.  1 1  paid  William  Mason  &  Co.  75%  of  my  indebtedness 
to  them  by  a  New  York  draft  for  $5100.     Jan.  8  they   sent  me 
goods  amounting  to  $  200.     Feb.  1  I  sent  them  a  check  in  full  of 
account.     What  was  the  amount  of  the  check  ? 


i  <? 


192  PERCENTAGE   AND   ITS  APPLICATIONS         [§§  524-527 

4.  Of  a  shipment  of  strawberries  10%  was  damaged,  and  the 
remainder,  which  were  sold  at  8^  per  quart,  brought  $146.88.     How 
many  quarts  were  there  in  the  shipment  ?  n^  ^  6  fl 

5.  A  grocer,  after  increasing  his  stock  by  goods  costing  $  6448, 
found  that  the  new  purchase  was  16%  of  the  old  stock  on  hand. 
What  was  the  value  of  his  old  stock  ? 

6.  From  his  bank  account  a  man  checked  out  ^  of  30%  of  his 
money,  and  after  depositing  $  750  he  found  that  he  had  in  the  bank 
105%  of  his  original  deposit.     How  much  had  he  in  the  bank  at 
first? 


7.  In  an  orchard  50%  of  the  trees  are  apple,  10%  peach,  20% 
pear,  and  the  remainder,  which  are  15  more  than  12J%  of  the  whole, 
are  plum.     How  many  trees  in  the  orchard  ? 

8.  A  floor  has  a  perimeter  of  84  ft.     If  the  width  is  75%  of  the 
length,  what  will  it  cost  to  paint  the  floor  at  22  ^  per  square  yard  ?  H  j 

9.  The  perimeter  of  the  ceiling  of  a  hall  is  130  ft.     If  the  widtlv^ 
of  the  hall  is  62i%  of  the  length  and  the  height  is  80%  of  the  width, 
and  360  sq.  ft.  are  deducted  for  openings,  what  will  it  cost  to  paint   - 
the  walls  and  ceiling  of  the  hall  at  15^  per  square  yard  ? 

10.   A  farmer  sold  18  bu.  2  pk.  and  1  qt.  of  tomatoes,  which  was 
5%  of  his  whole  crop.     How  many  bushels  of  tomatoes  did  he  raise  ? 

51  «  s\  fc  V  • 

525.  To  find  the  base,  the  amount  and  rate  of  increase  being  given. 

526.  Examples.     1.   What  number,  increased  by  87%  of  itself,  is 

equal  to  561  ? 

SOLUTION 

Represent  the  number  by  100%. 

87  %  =  the  increase. 

187  %  =  the  number,  increased  by  87  %  of  itself. 

661  —  the  number,  increased  by  87  %  of  itself. 

Therefore,  187%  of  the  number  =  561. 


100%,  or  the  number,  =  100  times  3,  or  300. 
Hence,  the  required  result  is  300. 

527.   Therefore  the  following  rule  : 
Divide  the  amount  by  1  plus  the  rate. 


§  527]  PERCENTAGE  193 

ORAL  EXERCISE 

What  number  increased  by : 

1.  10%  of  itself  is  77  ?     1  0  7.  125%  of  itself  is  675  ? 

&  331%  of  itself  is  36  ?  8.  40%  of  itself  is  280  ? 

8.  12|%  Of  itself  is  18  ?  £.  190%  of  itself  is  580  ? 

4.  20%  of  itself  is  240  ?  m  200%  of  itself  is  150  ? 

5.  40%  of  itself  is  28  ?  11.  300%  of  itself  is  1600  ? 

6.  50%  of  itself  is  63  ?  12.  6f  %  of  itself  is  160  ? 

WRITTEN  EXERCISE 

^.    I  sold  goods  for  $750  and  gained  25%.    What  did  they  cost? 

2.  I  gained  35%  by  selling  goods  for  $540.     What  was  the 
cost  ?       M 

3.  A  real  estate  dealer  gained  35%   by  selling   a   house  for 
$27,000.     What  was  its  cost? 

4.  I  sold  945  tubs  of  butter  for  $£103,  thereby  gaining  20%. 
How  much  did  the  butter  cost  per  tub  ?    // 

5.  A  drover  gained  18f  %  on  33  head  of  cattle  sold  for  $  1096.48. 
What  was  the  average  cost  per  head  ?  £~jf 

6.  The  attendance  of  pupils  at  school  during  May  was  954,  which 
was  6%  more  than  attended  during  April,  and  this  was  80%  more 
than   attended   during  February.      What  was   the   attendance   for 
February? 

7.  The  population  of  a  certain  town  has  increased  12^%  during 
the  past  three  years.     If  the  present  population  is  40,590,  what  was 
it  three  years  ago  ? 

8.  B  sold  a  farm  to  0  for  $  12,000,  thereby  gaming  20%.     What 
was  A's  cost  if  in  selling  the  same  farm  to  B  he  made  a  profit 
of  25%  ? 

9.  A  farmer's  wheat  crop  this  year  is  15J%  greater  than  last 
year's  crop.     What  was  last  year's  crop  if  in  the  two  years  he  raised 
1206|bu.?  ^tf>6 

10.   A  certain  number  plus  17%  of  itself,  plus  6%  of  3  times 
itself,  is  equal  to  270,135.     What  is  the  number  ? 
MOORE'S  COM.  AR.  — 13 


194  PERCENTAGE   AM)   ITS   APPLICATIONS         [§§528-530 

528.  To  find  the  base,  the  difference  and  rate   of  decrease  being 
given. 

529.  Example.     What  number,  decreased  by  35%  of  itself,  equals 
1300? 

SOLUTION 

Represent  the  number  by  100%. 

35  %  =  the  decrease. 

65  %  =  the  number  after  decrease. 

1300  =  the  number  after  decrease. 

Therefore,  65  %  =  1300. 

I%=  ft  of  1800,  or  20. 

100%  =  100  times  20,  or  2000,  the  required  result. 

530.  Therefore  the  following  rule : 

Divide,  the  difference  by  1  minus  the  rate. 

ORAL  EXERCISE 

What  number  diminished  by : 

1.  8%  of  itself  equals  184?  4.   25%  of  itself  equals  33  ? 

2.  16|%  of  itself  equals  55?  5.   50%  of  itself  equals  27  J  ?  \ 

3.  12|%  of  itself  equals  77?  &   f  of  itself  equals  339  ? 

7.  I  sold  goods  for  $  600  and  Ipst  40%.     What  did  they  cost  ? 

8.  Brown  deposited  $850  in  a  savings  bank,  which  was  15%    • 
less  than  that  deposited  by  his  son.     How  much  was  deposited  by 
both  of  them  ? 

9.  An  agent  earned  15%  less  in  May  than  he  did  in  June.     If  ^ 
he  earned  $  370  in  the  two  months,  how  much  did  he  earn  in  June  ?    ' 

WRITTEN  EXERCISE 

1.  A  boat  load  of  wheat  was  so  damaged  that  it  sold  for  $  8500, 
which  was  15%  less  than  its  original  value.     What  was  its  value 
before  it  was  damaged  ? 

2.  After  paying  37^%  of  his  debts  a  man  found  that  the  remain- 
der could  be  paid  with  $  13,025.     What  was  his  original  indebted- 
ness? 

3.  Smith  sold  two  horses  for  $1500  each,  gaining  25%  on  the 
first  and  losing  25%  on  the  second.     What  did  the  horses  cost  him  ? 


§  $30]  PERCENTAGE  195 

4.  A  liveryman  paid  $456  for  a  horse  and  carriage*     If  the  cost 
of  the  carriage  was  48%  less  than  the  cost  of  the  horse,  what  was 
the  cost  of  each  ? 

5.  A  merchant's  sales  for  Wednesday  were  50%    greater  than 
his  sales  for  Tuesday,  which  were  20%  less  than  his  sales  for  Mon- 
day.    If  the  sales  for  the  three  days  aggregated  $1500,  what  were 
the  sales  for  each  day  ? 

6.  In  selling  a  suit  of  clothes  for  $21.60  a  merchant  lost  20%. 
Find  the  asking  price  if  it  was  20%  above  cost. 

7.  A  man  bought  a  watch,  and  had  $  125  remaining.     He  then 
bought  another  watch  costing  twice  as  much  as  the  first,  and  still 
had  left  14^%  of  his  money.     How  much  money  had  he  at  first  ? 

8.  Divide  $1600'  between  A  and  B  so  that  B  shall  have  40% 
less  than  A. 

WRITTEN  REVIEW 

1.  A  benevolent  lady  gave  $  10,500  to  three  charities.     To  the 
first  she  gave  $2500,  to  the  second  $4500,  and  to  the  third  the 
remainder.     What  per  cent  did  each  receive  ? 

2.  25%  of  B's  money  equals  75%  of  A's.     How  much  has  A  if 
B  has  $900?     ^H 

3.  A  creditor,  after  collecting  33|%  of  a  claim,  lost  the  remain- 
der, which  was  $  3918,75.     What  sum  was  collected  ? 

v\     4-    A  has  185%  more  money  than  B.     How  much  has  each  if 
they  together  have  $  9625  ? 

5.  The  sum  paid  for  two  farms  was  $19,200.     37^%  of  the  sum 
.paid  for  one  equals  62 J%  of  the  sum  paid  for  the  other..    Find  the 

price  of  each. 

6.  If  a  gain  of  $4775  was  realized  on  a  business  at  the  end  of 
the  first  year,  and  a  loss  of  $3586.25  was  sustained  the  second  year, 
what  was  the  per  cent  of  net  gain  or  loss  for  the  two  years,  the 
investment  having  been  $63,400  ? 

7.  After  making  7  of  the  10  annual  payments  of  the  face  of  a 
mortgage  I  find  $  5850  to  be  still  unpaid.     How  many  dollars  have 
been  paid  ? 

8.  A  has  50%  more  money  than  B.     What  per  cent  has  B  less 
than  A  ? 


196  PERCENTAGE   AND   ITS  APPLICATIONS  [§  530 

9.  The  population  of  a  certain  city  is  238,375.  During  the  last 
three  years  the  population  has  increased  yearly  25%.  What  was  the 
population  three  years  ago ?  ^ 'V'V ^ C  ^\ 

10.  80%  of  a  mixture  of  vinegar  and  water  is  vinegar.     If  there 
were  10  gallons  more  of  vinegar,  the  mixture  would  be  85  %  vinegar. 
How  many  gallons  of  water  are  in  the  mixture  ? 

•\s 

11.  From  an  estate  the  widow  received  $  9250,  which  was  \ ;  the 
remainder  was  divided  among  three  children .  aged  respectively  15, 
12,  and  10  years,  and  they  share  in  proportion  to  their  age.   ,  What 
per  cent  of  the  estate  did  each  of  the  children  receive  ?  .Z  7  3*7 

12.  From  a  farm  containing  180  acres  120  square  rods,  50%  was 
sold  at  one  time  and  50%  of  the  remainder  at  another  time-    What 
per  cent  of  the  whole  then  remained  ? 

18.  After  drawing  25%  of  his  deposit  from  a  bank  to  pay  a  debt 
a  man  finds  that  he  has  left  in  the  bank  $  6756.25.  What  was  the 
amount  of  his  indebtedness,  and  how  much  had  he  in  the  bank 
before  drawing  the  check? 

14.  A.  man  sold  two  farms  for  $8000  each,  receiving  for  one  20% 
more  than  it  cost  and  for  the  other  20%  less  than  it  cost.     Did  he 
gain  or  lose  by  the  sale,  and  how  much  ? 

15.  During  the  first,  second,  and  third  years  a  manufacturer 
realized  gains  amounting  to  25%,  20%,  and  45%,  respectively,  of 
his  original  investment.     During  the  fourth  year  he  lost  $9000, 
which  was  25%  of  his  original  capital.     Find  his  net  gain  for  the 
four  years. 

16.  A  manufacturer's  capital  was  increased  during  the  first  year 
by  profits  equal  to  25%  of  his  original  investment ;  the  second  year 
by  profits  equal  to  20%  of  his  capital  at  the  beginning  of  that 
year;  and  the  third  year  it  was  diminished  by  a  loss  equal  to  25% 
of  the  capital  at  the  beginning  of  that  year.     If  his  profits  during 
the  three  years  exceeded  his  losses  by  $8000,  what  was  his  original 
investment  ? 

17.  In  settling  an  estate  an  executor  found  1\  %  of  it  to  be  invested 
in  telegraph  stock,  15%  in  railroad  stock,  37|%  in  United  States 
bonds,  $16,750  in  real  estate,  and  $7350  cash  in  bank.     Find  the 
total  value  of  the  estate. 


§  530]  PERCENTAGE  197 

18.  A  horse  is  worth  25%  more  than  a  carriage,  and  the  carriage 
is  worth  300%  of  the  harness.     If  the  horse  is  worth  $37.50  more  ' 
than  the  carriage,  what  is  the  value  of  each  ? 

19.  A  merchant  sold  2  horses  for  $  140  each.     On  one  he  gained 
25%  and  on  the  other  he  lost  28-f-%.     How  much  did  the  horses  cost 
him,  and  what  was  the  gain  or  loss  ? 

20.  Express  .00025  as  a  per  cent;  -^%  as  a  decimal;  36.42  as  a 
per  cent. 

21.  Find  3%  of  9  t.  7  cwt.  16  Ib. 

*i&&  I  paid  for  transportation  on  an  invoice  of  goods  $600.     I    ^/ 
later  sold  the  goods  at  a  profit  of  20%  on  the  full  cost,  receiving 
$9696.60.     What  was  the  first  cost  of  the  goods?     What  per  cent 
was  the  value  of  the  goods  increased  by  the  transportation  charges  ? 

28.   A  last  will  and  testament  provided  that  f  of  an  estate  dis- 
tributed should  go  to  the  widow  and  the  remainder  be  so  divided 
among  two  sons  and  a  daughter  that  the  elder  son  should  receive^.  if>>L 
10%  more  than  the  younger,  who  should  receive  25%  more  than  the?'   ^ 
daughter.     W^hat  amount  was  received  by  each,  the  estate  being 
valued  at  $58,000? 

24.  A  father  located  his  son  upon  a  farm,  expending  for  the 
farm,  stock,  utensils,   and   household   furniture   $19,512.50.     The 
stock  cost  twice  as  much  as  the  household  furniture,  which  cost 
75%  more  than  the  farm  utensils,  and  the  cost  of  the  farm  was 
140%  of  the  cost  of  the  stock.     How  much  was  invested  in  each  ? 

25.  A  creditor  agrees  to  receive  $  962.50  for  the  full  amount  of  a 
debt.    If  this  settlement  is  at  the  rate  t)f  25  f  on  the  dollar,  what  was 
the  original  amount  of  the  debt  ? 

26.  A  manufacturer  failing  in  business  finds  that  his  net  resources 
aggregate  $12,600.     If  he  can  pay  75  f  on  the  dollar  on  20%  of  his 
debts,  and  60^  on  the  dollar  on  the  remainder,  what  is  the  amount 
of  his  indebtedness. 

27.  Twice  ^  of  a  number  is  what  per  cent  of  3  times  J  of  it  ? 

28.  A  farm  is  composed  of  20%  more  grazing  than  grain  land, 
and  the  timber  is  1  of  the  area.     How  many  acres  of  each  are  there 
if  after  deducting  12  acres  for  lawn  and  garden  the  area  of  the  farm 
is  1860  acres  ? 


PERCENTAGE  AND  ITS  APPLICATIONS         [§§  530-537 

29.  A  man  withdrew  25%  of  his  bank  deposit  and  spent  20% 
of  the  money  to  pay  for  20%  of  his  indebtedness  to  Smith  &  Co.  If 
his  indebtedness  to  Smith  &  Co.  was  $1800,  what  was  the  original 
amount  of  his  bank  account  ? 

SO.  A  merchant  mixed  100  Ib.  of  coffee  at  25^  per  pound  with 
50  Ib.  at  30^  per  pound,  and  sold  the  mixture  for  40/.  What  per 
cent  of  profit  does  he  make  ? 

531.  Applications  of  percentage.    Percentage  is  applied  to  two 
general  classes  of  problems: 

1.  Those  in  which  time  is  not  an  element ;  as,  Commercial  Dis- 
counts, Gain  and  Loss,  Commission,  Insurance,  Taxes,  and  Customs, 
or  Duties. 

2.  Those  in  which  time  enters  as  an  element ;  as,  Interest,  Bank 
Discount,  Present  Worth  and  True  Discount,  Equation  of  Accounts, 
and  Exchange. 

COMMERCIAL  DISCOUNTS 

532.  Discount  is  an  allowance  made  for  the  payment  of  a  debt 
before  it  becomes  due. 

533.  Commercial  discounts  are  reductions  from  the  fixed  or  list 
prices  of  articles,  the  amount  of  a  bill  of  merchandise,  or  of  any 
other  obligation. 

534.  Commercial  discounts  embrace  trade  discounts,  time  dis- 
counts, and  cash  discounts. 

535.  Trade  discounts  are  reductions  from  the  fixed  or  list  prices 
of  articles. 

536.  Time  discounts  are  reductions  from  the  amount  of  a  bill  of 
merchandise  for  payment  within  a  definite  time. 

537.  Cash  discounts  are  reductions  made  for  the  immediate  pay- 
ment of  a  bill  of  merchandise  sold  on  time. 

Business  houses  usually  announce  their  terms  upon  their  billheads;  as 
••  Terms :  3  months,  or  5  %  off  for  cash  ; "  «« Terms  :  60  days,  or  3  %  discount  in 
10  days  ; "  etc.  When  bills  are  paid  before  maturity,  legal  interest  for  the 
remainder  of  the  time  is  usually  deducted. 


§§537-543]  COMMERCIAL   DISCOUNTS  199 

Trade  discounts  are  deducted  from  the  list  price  when  goods  are  billed. 
Time  discounts  are  deducted  when  a  bill  is  paid.  Cash  discounts  are  deducted 
from  the  amount  of  the  bill  when  the  sale  is  made. 

538.  It  is  customary  for  manufacturers,  jobbers,  and  wholesale 
dealers  to  have  fixed  price  lists  for  their  goods.     Trade  discounts 
are  usually  made  to  obviate  the  necessity  of  changing  these  price 
lists  from  time  to  time  as  the  market  changes.     As  the  market 
varies,  instead  of  changing  their  price  lists  or  issuing  new  cata- 
logues, merchants  raise  or  lower  their  rates  of  discount. 

539.  The  fluctuations  of  the  market  sometimes  give  rise  to  two 
or  more  discounts  known  as  a  discount  series.     If  two  or  more  dis- 
counts are  quoted,  the  first  denotes  a  discount  off  the  list  price,  the 
second  off  the  remainder,  and  so  on. 

540.  The  list  price  is  called  the  gross  price,  and  the  price  after 
the  discount  has  been  deducted,  the  net  price. 

541.  Commercial  discounts  are  usually  computed  by  the  rules  of 
percentage,  the  list  price  or  the  amount  of  the  bill  or  debt  corre- 
sponding to  the  base,  the  per  cent  of  discount  to  the  rate,  the  discount 
to  the  percentage,  and  the  net  price  or  net  amount  of  the  bill  or  debt 
to  the  difference. 

542.  To  find  the  net  selling  price,  the  list  price  and  discount  series 
being  given. 

543.  Examples.     1.  Find  the  net  amount  of  a  bill  of  $  450  after 

a  discount  of  33^%  is  made. 

SOLUTION 

$450  =  the  list  price. 

83i%,  or  £,  of  |450  =  $150,  the  discount  allowed. 

$450  —  $150  =  $300,  the  net  amount  of  the  bill. 

£  The  list  price  of  a  piano  is  $  800.  What  is  the  net  price  if  a 
discount  series  of  25%  and  20%  is  allowed  ? 

SOLUTION 

$800  =  the  list  price. 

25%,  or  £,  of  $800  =  $200,  the  first  discount. 
$800  -  $200  =  $600,  the  remainder  after  the  first  ditooimt 
20%,  or  $,  of  $600  =  $120,  the  second  discount. 
f  600  -  $  120  =  $480,  the  net  price. 


200  PERCENTAGE   AND   ITS  APPLICATIONS  [§544 

544.    Therefore  the  following  rule : 

Deduct  the  first  discount  from  the  list  price,  and  each 
subsequent  discount  from  each,  successive  remainder. 
The  last  remainder  is  the  net  selling  price. 

The  order  in  which  the  discounts  of  any  series  are  considered  is  not  material, 
a  series  of  25%,  15%,  and  10%  being  the  same  as  one  of  15%,  10%,  and  25%, 
or  10%,  25%,  and  15%. 

ORAL  EXERCISE 

1.  Find  the  net  cost  of  a  piece  of  glass  listed  at  $  3.60  and  dis- 
counted 25%. 

2.  A  merchant  sold  a  bill  of  goods  amounting  to  $  4.50  on  which 
he  allowed  20%  discount.     What  was  the  net  amount  of  the  bill  ? 

3.  Find  the  net  amount  of  a  bill  of  $  18,  the  discount  being  11^%. 

4.  What  is  the  net  amount  of  a  bill  of  $  450,  the  discounts  being 
33|%  and  20%  ? 

5.  A  piano  listed  at  $450  is  sold  less  33^%  and  10%.     What  is 
the  net  cost  to  the  purchaser  ? 

6.  Goods  listed  at  $27  are  sold  less  331%  and  16f  %.     What  is 
the  net  selling  price  ? 

WRITTEN  PROBLEMS 
Find  the  net  amount  of  the  following  bills : 

1.  $  1550  less  331%  and  20%/: '         8.  $3500  less  20%  and  14*  %. 

2.  $840  less  25%  and  10%.  9^         4.   $395  less  20%  and  20%.    i 

5.  A  wholesale  dealer  offers  cloth  at  $  2.40  per  yard  subject  to 
a  discount  of  25%,  20%,  and  5%.     How  many  yards  can  be  bought 
for  $492.48?  'SWA" 

6.  Find  the  net  price  of  2  tons  of  fence  wire  listed  at  3^  per 
pound  and  sold  20%  and  25%  off. 

7.  One  drummer  offers  to  sell  me  $1500  worth  of  iron  pipe  at  a 
discount  of  25%,  10%,  and  10%  ;  another  offers  to  sell  me  a  similar 

'quantity  of  pipe  for  the  same  amount  less  20%,  20%,  and  5%. 
Which  is  the  better  offer,  and  what  is  the  difference  expressed  in 
dollars  ? 

8.  Having   bought   $1500  worth  of  merchandise  at  20%  and 
25%  off,  I  sold  it  for  $1500  less  15%,  10%,  and  20%  off.     Did  I 
gain  or  lose,  and  how  much  ? 


544-540] 


COMMERCIAL   DISCOUNTS 


201 


9.  Books  purchased  at  25%  and  20%  off  from  the  list  price 
were  sold  at  the  list  price.  What  was  the  gain  per  cent?  What 
was  the  cost  of  a  shipment  which  sold  for  $700  ? 

10.  A  bill  of  hardware  is  sold  as  follows:  $25.50  at  20%  ;  $4.50 
at  20%  and  25%;  $153  at  33^%  and  10%;  $267.50,  net.  If  a 
further  discount  of  2%  is  allowed  for  immediate  payment,  what  is 
the  net  amount  of  the  bill  ? 

545.  To  find  the  net  amount  of  a  bill  to  render,  the  terms  and  dis- 
count series  being  given. 

546.  Ex-mple.     Bender  a  bill   for  the  following  transaction: 
Feb.  18,  1903,  E.  W,  Wells,  Medford,  Mass.,  bought  of  Baker,  Tay- 
lor &  Co.,  Boston,  Mass.:  1000  ft.  iron  pipe  at  25^  less  20%  and 


. 


Boston,  Mass.,. 


<Boaght  of  BAKER,  tAYLOR  &  CO. 


Terms;  <*30- 


-2-  *%  / 


204  PERCENTAGE   AND   ITS   APPLICATIONS         [§§  551-553 

551.  To  find,  mentally,  a  single  discount  equivalent  to  a  series  of 
two  discounts, 

From  the  sum  of  the  discounts  subtract  their  product,  and  the 
remainder  will  be  the  direct  discount. 

ORAL  EXERCISE 

By  inspection  find  a  single  rate  of  discount  equivalent  to  the  fol 
lowing  discount  series : 

1.  20%  and  10%.        11.   20%  and  12|%.      21.   10%  and  121%, 

2.  10%  and  10%.        12.   20%  and  20%,        22.  10%  and  6%. 

5.  25%  and  10%.  IS.  25%  and  25%.  23.  15%  and  6%. 
4.  30%  and  10%.  14.  5%  and  5%.  24.  25%  and  8%. 
d.  20%  and  5%.  15.  60%  and  25%.  25.  33i%  and  6%. 

6.  10%  and  5%.  16.  25%  and  20%.  26.  40%  and  12^%. 

7.  40%  and  10%.  17.  30%  and  20%.  07.  1H%  and  18%. 

8.  25%andS3j%.  18.  10%  and  30%.  28.  81%  and  24%. 
P.  20%  and  33^%.  19.  35%  and  20%.  29.  16f  %  and  12%. 

10.   15%  and  10%.        20.  20%  and  15%.       £0.   14*%  and  35%. 

552.  When  a  discount  series  consists  of  three  rates  of  discount, 
combine  the  first  two,  and  then  the  result  obtained  and  the  third. 

553.  Example.     Find  a  single  rate  of  discount  equivalent  to  a 
series  of  25%,  20%,  and  10%, 

SOLUTION.  Combine  the  first  two  by  saying  or  thinking,  25%  -f  20%  -  6% 
=  40%,  or  the  single  discount  equivalent  to  the  series  of  25%  and  20%. 

40  %  +  10  %  -  4  %  =  46%,  or  the  single  rate  of  discount  equivalent  to  the  series 
25%,  20%,  and  10%. 

ORAL  EXERCISE 

By  inspection  find  a  single  rate  of  discount  equivalent  to  the 
following  series: 

1.  20%,  25%,  and  20%.  6.  20%,  12£%,  and  10%. 

2.  10%,  10%,  and  20%  6.  20%,  20%,  and  10%. 

3.  20%,  5%,  and  10%.  7.  20%,  15%,  and  10%. 

4.  30%,  20%,  and  10%.  8.  25%,  33i%,  and  10% 


§§  553-556J  COMMERCIAL  DISCOUNTS  205 

WRITTEN  EXERCISE 

1.  From  a  list  price  I  discounted  30%,  25%,  and  20%.     What 
per  cent  better  for  the  purchaser  would  a  single  discount  of  70% 
have  been?  \^a\0 

2.  A  merchant  purchasing  a  bill  of  goods  was  allowed  discounts 
from  the  list  price  of  15%,  10%,  10%,  and  6%.     If  the  total  dis- 
count allowed  was  $352.81,  what  must  have  been  the  asking  price 

of  the  goods  ?    - 
f 

3.  Goods  were  sold  25%,  35%,  20%,  and  15%  off.     If  the  total 
discount  allowed  was  $  334.25,  what  must  have  been  the  net  selling 
price  of  the  goods  ? 

4-   The  net  amount  of  a  bill  was  $1080  and  the  discounts  were 
20%,  25%,  and  10%.     Find  the  amount  of  the  discount  allowed. 

5.  Goods  sold  on  account  30  days,  2%  10  days,  are  paid  for  on 
the  date  of  sale.     What  must  have  been  the  gross  amount  of  a  bill 
of  goods  on  which  $  588  cash  was  paid  if  the  discount  series  was 
25%  and  20%? 

6.  If  the  list  price  on  an  article  is  25%  advance  on  the  cost, 
what  other  per  cent  of  discount  than  10%  must  be  allowed  to  net 
10%  gain  by  the  sale? 

554.  To  find  the  net  cost  of  goods,  the  list  price  and  discount 
series  being  given. 

555.  Example.     A  piano  is  listed  at  $  450  and  the  discounts  are 
20%  and  25%.     Find  the  net  cost  to  the  purchaser. 

100%  —  40%  =  60%.  SOLUTION.      Mentally  determine  a  single 

f  ft  4^0  —  ft  2^0         rate  °*  Discount  equivalent  to  a  series  of  25% 

OI  *JP  *±O\J  —  <JP  ^  I  U.  ,_j    on  n/        rpu;«   5^,   t^-.-.^A    *^  i^    A(\nt        •D/i-m.^ 


- 

'  and  20%.  This  is  found  to  be  40%.  Repre- 
sent the  gross  cost  by  100%.  100%  minus  40%,  the  direct  discount,  leaves  60%, 
the  net  cost  to  the  purchaser.  60%  of  the  gross  cost  is  $270,  or  the  net  cost  to 
the  purchaser. 

556.   Hence  the  following  rule : 

Multiply  the  list  price  by  the  difference  per  cent,  and  the  product 
will  be  the  net  cost. 

When  It  is  not  desirable  to  show  the  discounts  on  a  bill,  the  above  method 
will  be  found  the  most  practicable  for  finding  the  net  cost. 


206  PERCENTAGE   AND   ITS   APPLICATIONS         [§§556-500 

ORAL  EXERCISE 

Find  the  net  cost  of  articles  listed  at : 

L   $200  less  20%  and  10%.         4.    $200  less  25%  and  8%. 

2.  $300  less  20%  and  15%.         5.    $1000  less  30%  and  5%. 

3.  $600  less  25%  and  20%.         6.    $18.50  less  25%  and  20%. 

WRITTEN  EXERCISE 

1.  Find  the  net  cost  of  articles  listed  as  follows :  lead  pipe,  65  ^, 
discount,  40%  and  10%;  iron  pipe,  30^,  discount,  45%  and  20%; 
bath  tubs,  $12,  $10,  and  $8,  discount,  20%  and  10%. 

2.  Five  pianos  listed  at  $425  each  were  sold  at  a  discount  of 
20%  and  25%.     If  the  freight  was  $27.50  and  the  dray  age  $15, 
what  was  the  net  amount  of  the  bill  ? 

8.   Find  the  net  cost  of  an  organ  listed  at  $175,  subject  to  a 
discount  of  20%  and  10%. 

4.  Which  is  the  cheaper,  and  how  much,  on  a  bill  of  $500,  a 
discount  series  of  60%,  20%,  and  10%,  or  a  discount  series  "of  50%, 
40%,  and  10%  ? 

5.  To  the  net  cost  of  an  article  $  30  was  added  for  freight,  making 
the  total  net  cost  $  120.     What  was  the  list  price,  the  rates  of  dis- 
count being  50%  and  5%  ? 

GAIN  AND  LOSS 

557.  The  gains  and  losses  arising  from  business  transactions  are 
frequently  computed  as  percentages  of  the  cost. 

558.  The  first  cost  of  goods  is  called  the  prime  cost.     The  prime 
cost,  increased  by  all  direct  outlays  incident  to  the  purchase  and 
holding  of  the  goods  to  the  date  of  sale,  such  as  packing,  freight, 
cartage,  storage,  commission,  etc.,  is  called  the  gross  or  full  cost. 

559.  The  actual  amount  arising  from  the  sale  of  goods  is  called 
the  gross  selling   price.     The  gross  selling  price,  less  all   charges 
incident  to  the  sale  of  the  goods,  is  called  the  net  selling  price. 

560.  The  difference  between  the  net  selling  price  and  the  gross 
cost  of  the  goods  is  the  gain  or  loss,  —  a  gain  if  the  selling  price  is 
the  larger,  and  a  loss  if  the  gross  cost  is  the  larger. 


§§  561-562]  GAIN   AND   LOSS  207 

561.  In  ascertaining  the  gain  or  loss  the  operations  are  usually 
performed  by  the  rules  of  percentage,  the  gross  cost  corresponding  to 
the  base,  the  per  cent  of  gain  or  loss  to  the  rate,  the  gain  or  loss  to  the 
percentage,  the  net  selling  price,  if  at  a  gain,  to  the  amount,  and  the 
net  selling  price,  if  at  a  loss,  to  the  difference. 

562.  To  find  the  gain  or  loss,  the  cost  and  per  cent  of  gain  or  loss 
being  given. 

ORAL  EXERCISE 

By  inspection  find  the  gain  or  loss  in  each  of  the  following 
problems  : 


Per  Cent  of 
Co8t              Gain 

Per  Cent  of 
Cost              Gain 

Per  Cent  of 
Cost            Loss 

1. 

$3600 

25 

6. 

$280 

14f 

11. 

$2500 

36 

2. 

$2500 

36 

7. 

$1500 

66| 

12. 

$1250 

16 

3. 

$  12,500 

16 

8. 

$960 

33J 

IS. 

$480 

37J 

4- 

$250 

24 

9. 

$420 

"  28| 

14. 

$3200 

3H 

5. 

$3750 

32 

•  10. 

$500 

25 

15. 

$630 

18f 

16-30.   Find  the  selling  price  in  each  of  the  above  problems. 

WRITTEN  EXERCISE 

1.  A  grocer  bought  10  bbl.  of  sugar,  each  weighing  330  lb.,  at 
per  pound,  and  sold  them  so  as  to  gain  16-f  %.     Find  the  gain 

and  the  selling  priced  ^*f.'j 

2.  A  stock  of  goods  consisting  of  $25,000  worth  of  groceries 
was  sold  at  a  loss  of  12-1-  %,  and  15%  of  the  selling  price  was  in 
uncollectible  accounts.     What  was  the  total  loss  sustained  ?  tfLyA 

3.  A  produce  dealer  paid  $  320  for  apples,  $  90  for  onions,  and 
$  120  for  potatoes.     He  sold  the  apples  at  a  gain  of  25%,  the  onions 
at  cost,  and  the  potatoes  at  95%  of  their  cost.     Did  he  gain  or  lose, 
and  how  much  ? 

4-  A  man  bought  three  horses,  paying  respectively  $  240,  $  300, 
and  $  500.  He  sold  the  first  at  125%  of  its  cost,  the  second  at  a  loss 
of  10%,  and  the  third  at  a  gain  of  15%.  Did  he  gain  or  lose,  and 
how  much  ? 

5.  A  dry  goods  merchant  bought  a  bill  of  goods  amounting  to 
$175.  He  sold  14f  %  of  the  bill  and  realized  a  gain  equal  to  50% 
of  the  cost  of  the  whole  bill.  If  the  remainder  of  the  stock  was  sold 
for  $  100,  what  was  the  gain  or  loss  ?  <n  j"0 


208  PERCENTAGE   AND   ITS   APPLICATIONS  [§563 

563.  To  find  the  rate  of  gain  or  loss,  the  cost  and  gain  or  loss  being 
given. 

ORAL  EXERCISE 

By  inspection  find  the  per  cent  of  gain  or  loss  in  each  of  the 
following  problems : 

Cost  Gain  Cost        Gain  Cost       Loss  Cost          Loss 

1.  $125  f  12.50    6.  $200  $400       9.  $1.60  20^      13.  $25     $6.25 

2.  $150  $7.50       6.  $240  80^       10.  $3.60  40^      14.  $92     $23 

3.  $380  $76         7.  $100  $250     11.  $1.35  45^     15.  $85     $17 

4.  $2       $1  8.  $250  $100     12.  $190    $76     16.  $420  $70 
17-32.    Find  the  selling  price  in  each  of  the  above  problems. 

33.  What  per  cent  is  gained  by  selling  an  article  for  twice  its 
cost  ?  Three  and  one-fifth  times  its  cost  ? 

34-  A  speculator  bought  a  quantity  of  wheat  at  80^  per  bushel 
and  sold  it  at  $  1  per  bushel.  How  many  bushels  did  he  buy  if  his 
gain  was  $20?  /  Ft 

35.  If  a  merchant  sells  3  Ib.  of  sugar  for  what  4  Ib.  cost  him, 
what  is  his  gain  per  cent  ? 

WRITTEN  EXERCISE 

1.  Wheat  bought  at  85^  per  bushel  is  sold  at  $1.05  per  bushel. 
How  many  bushels  must  be  handled  to  realize  a  profit  of  $400  ? 

2.  If  I  bought  handkerchiefs  at  $3.25  per  dozen  and  retailed 
them  at  35^  each,  what  was  my  gain  per  cent  ? 

3.  A  coal  dealer  buys  his  coal  at  the  mines  by  the  long  ton.    If 
he  sells  at  an  advance  of  33^%  on  the  cost,  and  uses  the  short  ton 
weight,  what  is  his  gain  per  cent  ? 

4.  If  |  of  an  article  is  sold  for  what  }  cost,  what  is  the  loss  per 
cent? 

5.  If  \  of  an  article  is  sold  for  what  \  cost,  what  is  the  loss  per 
cent? 

0.    If  ^  of  an  article  is  sold  for  what  \  cost,  what  is  the  gain  per 
cent? 

7.   Paper  bought  at  $2.70  per  ream  is  retailed  at  1^  a  sheet. 
What  is  the  per  cent  of  gain  ? 


§§  563-564]  GAIN  AND   LOSS  209 

8.  A  merchant  bought  a  stock  of  goods  amounting  to  $  8500,  and 
after  disposing  of  a  part  of  it  for  $7500  he  took  account  of  the 
stock  remaining  unsold  and  found  that  at  cost  prices  it  was  worth 
$2700.  What  was  the  per  cent  of  gain  on  the  sales  ?  f\ri  M  ^> 

•Avv 

564.  To  find  the  cost,  the  gain  or  loss  and  the  per  cent  of  gain  or 
loss  being  given. 

ORAL  EXERCISE 

Find  the  cost  if : 

1.  25  %  loss  =  $  30.        4.  %  %  loss  =  $  100.         7.  15  %  gain  =  $  150. 

2.  20 %  gain  =  $  1.50.     6.  \ %  gain  =  $  30.  8.  14$%  gain  =  $  12. 

3.  30%  loss  =  $2.10.     6.  125%  gain  =  $3.75.     9.  22%  gain  =  $880. 
10-18.   Find  the  selling  price  in  each  of  the  above  problems. 

19.  What  must  have  been  the  cost  of  a  stock  of  goods  if  the 
owner,  by  selling  at  a  gain  of  12%%,  received  $450  more  than  the 
cost? 

WRITTEN   EXERCISE 

1.  A  dealer  sold  35%  of  a  purchase  of  leather  at  14f  %  gain  and 
the  remainder  at  5%  loss.     If  his  net  gain  was  $87.50,  what  must 
have  been  the  cost?  ~ 

2.  A  merchant  bought  goods  and  paid  freight  on  them  equal  to 
12%  of  their  cost.    He  then  sold  them  at  6J%  profit  on  the  full  cost 
of  the  goods,  receiving  60%  of  the  price  in  cash,  and  a  note  for  the 
remainder.    If  the  amount  of  the  note  was  $1309,  what  was  the  first 
cost  of  the  goods  ?  f  1.  f&t 

3.  A  dry  goods  merchant's  gain  in  business  for  four  years  aggre- 
gated 50%  of  his  capital.     If  his  gain  was  $5000  and  he  withdrew 
it  and  his  capital  and  invested  the  total  in  a  farm,  consisting  of  375  A., 
what  was  the  price  paid  per  acre  ?*  t- 

4.  Having  bought  a  house  of  A  at  12%%  less  than  it  cost  him,  I 
spent  $430  for  repairs  and  sold  it  for  $7293,  thereby  gaining  10% 
on  my  investment.     How  much  did  the  house  cost  A  ? 

5.  A  man  sold  a  horse  at  33^%  profit.     He  put  with  the  sum 
received  $50  and  bought  a  piano,  which  he  sold  at  20%  gain.    If  his 
total  gain  was  $  100,  what  was  the  cost  of  the  horse  ? 

MOORE'S  COM.  AR.  — 14 


210  PERCENTAGE   AND   ITS  APPLICATIONS  [§565 

565.  To  find  the  cost,  the  selling  price  and  the  per  cent  of  gain  or 
loss  being  given. 

ORAL  EXERCISE 

Find  the  cost  when  the  selling  price  at : 

1.  5  %  gain  =  $  105.       4.  125  %  gain  =  $  225.  7.  140  %  gain  =  $  480. 

2.  20%  gain  =  $240.     5.  12^%  loss  =  $140.  8.  14f%  loss  =  $2400. 

3.  16|%gain  =  $14.     6.  33^%  loss  =  $360.  9.  111%  loss  =  $3200. 

10-18.   Find  the  gain  or  loss  in  each  of  the  above  problems. 

19.  A  fruit  dealer,  after  losing  12^%  of  his  apples  by  frost,  had 
150  barrels  left.  If  he  bought  the  apples  at  $  2  per  barrel  and  sold 
at  $  3,  what  was  his  gain  ? 

WRITTEN  EXERCISE 

1.  I  sold  a  house  to  B  at  10%  profit.     B  sold  it  to  C,  gaining 
15%,  and  C,  by  selling  it  to  D  for  $15,939,  gained  20%  on  his  pur- 
chase.    How  much  did  the  house  cost  me  ? 

2.  I  sold  two  watches  at  the  same  price.     On  one  I  gained  25%, 
and  on  the  other  I  lost  25%.     If  my  total  loss  was  $10,  what  was 
the  cost  of  each  ? 

3.  A  sold  a  stock  of  silks  to  B  at  a  gain  of  25% ;  B  sold  the  same 
stock  to  C  at  a  gain  of  10%..    If  C's  cost  was  $375  more  than  A's, 
what  did  the  silks  cost  A  ? 

WRITTEN  REVIEW 

1.  What  amount  of  money  must   an   attorney  collect  in  order 
that  he  may  pay  over  to  his  client  $1700  and  retain  15%  for  his 
services  ?  *•< 

2.  In  selling  an  article  for  $162  an  art  dealer  cleared  12£%.    At 


what  per  cent  above  cost  was  it  marked  if  the  asking  price  was  $176? 

3.  A  man  bought  a  quantity  of  apples  at  $2  per  barrel.     He 
sold  \  of  them  at  $3  per  barrel,  -|  of  the  remainder  at  $3.25  per  bar- 
rel, and  the  remainder,  750  bbl.,  at  $2.50  per  barrel.     What  was  his 
gain? 

4.  By  selling  apples  at  $2.50  per  barrel  I  gained  $200.     Had  I 
sold  them  at  $  2.75  per  barrel  my  rate  of  gain  would  have  been  37 ',  %. 
How  many  barrels  did  I  sell  ?     -^ 


§565]  GAIN  AND   LOSS  211 

5.  A  dealer  bought  wheat  at  90  ^  per  bushel.    He  sold  f  of  it  at 
33 \°/o  gain  and  the  remainder  at  a  loss  of  $25.     If  his  gain  on  the 
whole  transaction  was  22f  %,  how  many  bushels  of  wheat  did  he  buy? 

6.  What  per  cent  of  gain  must  be  realized  on  an  engine  costing 
$  1928  in  order  that  it  may  be  sold  for  $  2410  ?    t  6"  % 

7.  A  produce  dealer  bought  24,000  Ib.  of  wheat  for  $360.    He 
sold  it  at  $1.05  per  bushel.     What  was  his  gain  per  cent  ?    J&  % 

8.  A  compromised  with  an  insolvent  debtor  at  the  rate  of  50^ 
on  a  dollar.     To  obtain  an  immediate  payment  he  allowed  a  further 
discount  of  5%.     What  was  the  amount  of  his  claim,  his  total  loss 
having  been  $  10,505.25  ?        \s>9  Dt 

9.  An  article  marked  to  gain  621%  is  sold  less  25%  and  20%. 
If  a  collector  was  afterwards  paid  20%  for  collecting  the  account, 
what  was  the  gain  or  loss  per  cent  ? 

10.  An  article  marked  25%  above  cost  is  sold  at  a  discount  of 
16|%.     Jf  the  gain  is  $25,  what  is  the  selling  price  of  the  article  ? 

11.  If  I  made  a  profit  of  16|%  by  selling  a  horse  at  $7,50 
above  cost,  how  much  should  I  have  received  above  cost  to  realize 
a  profit  of  25%  ? 

12.  What  per  cent  is  gained  by  buying  pork  at  $17.50  per  bar- 
rel, and  retailing  it  at  12  ^  per  pound  ? 

IS.  Having  bought  75  barrels  of  apples  for  $  187.50,  I  sold  them 
at  a  loss  of  20%.  How  much  did  I  receive  per  barrel  ? 

14.  I  lost  25%  of  a  consignment  of  berries.  At  what  per  cent 
of  profit  must  the  remainder  be  sold  in  order  that  I  may  gain  10  % 
on  the  whole  ? 


15.  A  Texas  farm  of  160  acres  was  bought  at  $15  per  acre; 
$  354  were  paid  for  fencing,  $  480  for  breaking,  $  626  for  a  house, 
and  $220  for  a  barn.     At  what  price  per  acre  must  it  be  sold  to 
realize  a  net  profit  of  25%  on  the  investment  ? 

16.  If  25%  of  the  selling  price  is   gain,  what  is  the  gain  per 
cent? 

17.  I  sell  |  of  a  stock  of  goods  for  $27,  thereby  losing  20%. 
For  what  must  I  sell  the  remainder  to  make  a  profit  of  20%  on  the 
whole  ? 


212  PERCENTAGE   AND   ITS  APPLICATIONS  [§565 

18.  A  banker  bought  a  mortgage  at  7\°/o  less  than  its  face  value, 
and  sold  it  for  3%  more  than  its  face  value,  thereby  gaining  $981.75. 
What  was  the  face  value  of  the  mortgage  ? 

19.  If  I  sell  J  of  an  acre  of  land  for  what  |-  of  it  cost,  what  will 
be  my  gain  or  loss  per  cent?   |U*M»  J^ 

20.  B  and  C  each  invested  an  equal  amount  of  money  in  busi- 
ness; B  gained  121%  on  his  investment,  and  C  lost  $5275;  C's 
money  was  then  42%  of  B's.     How  many  dollars  did  each  invest  ? 

21.  A   manufacturing   company's  per  cent  of  gain   on  a  self- 
binder  was  25%  less  than  that  of  the  general  agent;  the  general 
agent's  profit  was  20%,  he   thereby  gaining   $25.30.     What   did 
it  cost  to  make  the  machine? 

22.  For  what  must  hay  be  sold  per  ton  to  gain  16f  %,  if,  by  sell- 
ing it  at  $  18  per  ton,  there  is  a  gain  of  25%  ?    \  ^  .  J  0 

V  28.  A  stock  of  goods  is  marked  221%  advance  on  cost,  but 
becoming  damaged,  is  sold  at  20%  discount  on  the  marked  price, 
whereby  a  loss  of  $  1186.40  is  sustained.  What  was  the  cost  of  the 
goods  ? 

24.  Of  a  cargo  of  8000  bushels  of  oats,  costing  35  ^  per  bushel, 
25%  was  destroyed  by  fire.     What  per  cent  will  be  gained  or  lost  if 
the  remainder  of  the  oats  is  sold  at  45  ^  per  bushel  ?    ' 

25.  A  grocer  bought  200  quarts  of  berries  at  11  \$  per  quart, 
and  150  quarts  of  cherries  at  6 \  $  per  quart.     Having  sold  the  cher- 
ries at  a  loss  of  30%,  for  how  much  per  quart  must  he  sell  the 
berries  to  gain  15%  on  the  whole? 

26.  Having  bought  48  pounds  of  coffee  at  the  rate  of  3J-  pounds 
for  91  ^  and  84  pounds  more  at  the  rate  of  7  pounds  for  $  1.26,  I 
sold  the  lot  at  the  rate  of  9  pounds  for  $1.53.     What  was  my  per 
cent  of  gain  or  loss  ?    |  £  x^%   d^r&O 

27.  Having  paid  a  retailer  $138.60  for  a  set  of  furniture,  I 
ascertain  that  by  selling  to  me  he  gained  12|%,  that  the  whole- 
saler of  whom  he  bought  gained  10%,  that  the  jobber  by  selling  to 
the  wholesaler  gained  16 f  %,  and  that  the  manufacturer  sold  to  the 
jobber  at  20%  above  its  first  cost.     How  much  more  than  its  first 
cost  did  I  pay  ? 


§565]  GAIN   AND   LOSS  213 

28.  If  I  pay  $3.20  for  20  gallons  of  vinegar,  how  many  gallons  of 
water  must  be  added  that  40%  profit  may  be  realized  by  selling  it 
at  15^  per  gallon? 

29.  A  merchandise  account  shows  that  the  cost  of  a  stock  of 
goods  was  $15,000,  that  the  sales  to  date  aggregate  $12,000,  and 
that  the  goods  on  hand,  estimated  at  cost  prices,  amount  to  $  4500. 
Find  the  per  cent  of  gain  or  loss  on  the  sales. 

30.  A  sold  a  horse  to  B  and  gained  20% ;  B  sold  it  to  C  and 
gained  25%.     If  the  average  gain  was  $50,  what  was  C's  cost  ? 

81.  A  grocer  buys  10  barrels  of  apples,  each  barrel  containing  2-J- 
bushels  at  $2  per  barrel.  If  the  loss  by  decay  amounts  to  20%,  at 
what  price  per  peck  must  he  retail  them  in  order  to  clear  20%  ? 

32.  A  grocer  mixes  10  pounds  of  tea  costing  36  ^  per  pound  with 
8  pounds  costing  45  f  per  pound.  At  what  price  must  he  sell  the 
mixture  to  gain  25%  upon  his  outlay  ? 

S3.  A  stock  of-  imported  silks  bought  for  £200  was  sold  for 
$  1167.96.  What  was  the  gain  per  cent  ? 

34.  What  per  cent  is  gained  in  buying  coal  at  $4.50  for  a  long 
ton  and  retailing  it  at  $  6  a  short  ton  ? 

35.  What  per  cent  is  gained  on  quinine  costing  $2.90  an  ounce 
and  sold  at  2  $  a  grain  ? 

36.  A  merchant  buys  hardware  at  25%  and  20%  off  the  list 
prices,  and  sells  at  20%  and  10%  off  the  list  prices.     What  per 
cent  of  gain  does  he  realize  ? 

87.  A  manufacturer  sells  at  20%  and  10%  off  the  list  prices. 
His  blundering  clerk,  in  making  out  the  bill  for  the  goods,  deducted 
.30%.  If  the  discount  deducted  was  $450,  what  should  have  been 
the  net  amount  of  the  bill  ?  If  the  mistake  passed  unnoticed,  what 
was  the  buyer's  gain  per  cent  on  the  transaction  ? 

38.  A  dealer  in  agricultural  implements  marked  a  self-binder 
at  an  advance  of  25%  on  the  cost.  In  order  to  collect  an  account, 
he  had  to  pay  an  attorney  10%  of  the  amount  of  the  debt.  If  the 
sale  netted  him  a  gain  of  $25,  what  was  the  selling  price  of  the 
binder  ? 


214  PERCENTAGE   AND   ITS  APPLICATIONS         [§§  565-569 


89.   A  retailer  buys  collars  at  the  rate  of  $  1  a  dozen,  and 
at  the  rate  of  2  for  25^.     What  is  the  per  cent  of  gain  ?  *C^)       0 

40.  A  manufacturer  sold  a  retailer  a  piano  and  gained  20%. 
The  retailer  compromised  with  his  creditors,  paying  75^  on  the 
dollar.     What  was  the  manufacturer's  per  cent  of  loss  on  the  trans- 
action if  he  discounted  the  amount  which  he  could  legally  collect 
o%  for  immediate  payment? 

41.  An  insolvent  debtor  pays  his  creditors  37^  on  the  dollar. 
If  his  creditors  receive  $  3515,  what  is  their  joint  loss  ? 

42.  A  manufacturer  sells  to  a  wholesaler  at   20%    gain;   the 
wholesaler  to  the  retailer  at  25%   gain;   and  the  retailer  to  the 
consumer  at  60%  gain.     Find  the  cost  to  the  manufacturer  of  an 
article  for  which  the  consumer  pays  $  2.40  more  than  twice  the  cost 
to  manufacture. 

MARKING  GOODS 

566.  In  marking  goods,  it  is  customary  for  merchants  to  use  a 
word,  phrase,  or  an  arbitrary  arrangement  of  characters  called  a  key 
to  represent  the  ten  Arabic  numerals.     In  this  way  the  cost  and 
selling  price  may  be  written  on  an  article  and  yet  be  unintelligible 
to  all  except  those  who  know  the  key. 

567.  If  letters   are  used,  any  word  or  phrase  containing  ten 
different  letters  may  be  selected.     If  characters  are  used,  any  ten 
different  arbitrary  characters  may  be  selected. 

568.  Merchants  generally  use  two  different  keys,  one  to  repre- 
sent the  cost  and  one  the  selling  price. 

569.  To  avoid  the  repetition  of  a  letter,  and  to  make  the  key 
more  valuable  as  a  private  mark,  one  or  two  extra  letters  called 
repeaters  are   used  to   indicate   letters  which  would  otherwise  be 
repeated. 

To  illustrate  the  method  of  marking  goods,  take  the  following  keys  : 

Cost  Mark  Selling  Price  Mark 

SEZIBOHTUA          HTIMSKCALB 

1234667890  1284667890 

Repeaters  :  G  and  F.  Repeaters  :  W  and  O. 


§§  569-573]  MARKING   GOODS  215 

It  will  be  observed  that  the  words  authorizes  and  blacksmith,  spelled  back- 
wards, are  used  to  represent  the  cost  and  selling  price  respectively  ;  also  that 
the  repeaters  in  both  cases  are  mere  arbitrary  letters. 

The  cost  and  selling  price  are  generally  written  one  above  and  the  other 
below  a  line  on  a  tag,  or  upon  a  paster  or  box.  • 

Thus,  if  the  above  keys  are  used,  * — ^ — ,  on  a  dozen  of  gloves,  would  be 

$  C.oB 

understood  to  mean  that  the  cost  was  $6.00  per  dozen,  and  that  the  selling  price 
is  $7.50  per  dozen. 

WRITTEN  EXERCISE 

Using  the  keys  given  in  569,  write  the  cost  and  selling  price  of 
articles  costing : 

1.  $2.50  and  selling  at  20%  gain ;  $2.40  and  selling  at  25%  gain. 

2.  $1.80  and  selling  at  33£%  gain ;  $4.20  and  selling  at  16f  %  gain. 

3.  18^  and  selling  at  16f  %  gain ;  $27  and  selling  at  30%  loss. 

4.  $4.26  and  selling  at  16f  %  gain;  $3.60  and  selling  at  12^%  gain. 

5.  $425  and  selling  at  20%  gain ;  $24.90  and  selling  at  33£%  gain. 

6.  $16  and  selling  at  31^%  gain;  $2.40  and  selling  at  37£%  gain. 

570.  While  the  unit  of  measure  varies  with  the  quantities  and 
qualities  offered  for  sale,  a  large  number  of  manufactured  products 
are  sold  by  the  dozen.     Jobbers  and  wholesalers  buy  a  great  many 
articles  by  the  dozen.     Retailers  buy  a  great  many  articles  by  the 
dozen,  but  usually  sell  them  by  the  piece. 

571.  To  find  the  cost  price  of  an  article,  the  cost  price  of  a  dozen 
being  given. 

572.  Example.     If  1  doz.  hats  cost  $  25,  what  will  1  hat  cost  ? 

SOLUTION.  Since  a  dozen  hats  cost  $25,  1  hat  will  cost  T^  of  $25,  or  $2^, 
which  is  equal  to  .$2.08$  or  $2.08. 

It  is  just  as  easy  to  divide  a  number  by  12  as  it  is  by  any  number  of  one 
digit.  Hence,  in  dividing  by  12,  use  the  short  division  method.  After  dividing 
the  figures  in  the  dividend,  consider  the  remainder  as  twelfths  of  a  number,  and 
mentally  reduce  it  to  an  approximate  decimal. 

573.  The  following  table   shows   the    decimal   values    of    the 
twelfths  which  may  remain  after  dividing  a  number  by  12. 


216 


PERCENTAGE   AND   ITS  APPLICATIONS 


573-676 


TABLE  OF  TWELFTHS 


TWELFTHS 

SIMPLEST 
FORM 

DECIMAL  VALUE 

TWELFTHS 

SIMPLEST 
FOBM 

DECIMAL  VALUE 

A 

$.08$ 

A 

$.58| 

A 

i 

.16| 

A 

1 

.66| 

A 

i 

.25 

A 

I 

.75 

A 

i 

.33$ 

u 

i 

.88} 

A 

•41f 

H 

-91| 

A 

* 

.5 

H 

i 

1. 

Familiarity  with  the  above  table  will  give  facility  in  dividing  numbers 
by  12. 

ORAL  EXERCISE 

By  inspection  find  the  cost  of  one  article  when  billed  by  the 
dozen  as  follows : 

1.  Shoes  at  $18.60.    5.   Ties  at  $9.  9.   Hose  at  $3.90. 

2.  Boots  at  $42.          6.   Coats  at  $116.        10.   Gloves  at  $  3.90. 
8.  Hose  at  $6.60.        7.   Scarf s  at  $  1.32.      11.  Hose  at  $5. 

4.   Hats  at  $27.  8.   Shirts  at  $11.60.    12.   Caps  at  $  16.90. 

574.  To  find  the  selling  price  of  an  article,  the  cost  per  dozen  and 
the  rate  per  cent  of  gair  being  given. 

575.  Example.     Find  the  selling  price  of  a  pair  of  gloves  so  as 
to  net  a  profit  of  33^%,  the  cost  per  dozen  being  $7.50. 

£X  $7.50  =  $2.50. 
$7.50 +  $2.50  =  $10. 


SOLUTION.     Since  33  J%,  or  $,  of  the  cost 
is  profit,  the  selling  price  per  dozen  will  be  \ 
more  than  $  7.50,  or  $  10.    $  10  divided  by  12 
$  10  -J-  12  =  $  f  =  83  £        equals  83^,  or  the  selling  price  per  pair. 


576.    Therefore  the  following  rule : 

Find  the  selling  price  of  one  dozen  "by  adding  to  the  cost 
of  one  dozen  such  a  part  of  the  cost  as  the  rate  per  cent  of 
gain  is  a  part  of  100%. 

Find  the  cost  per  article  by  dividing  the  cost  per  dozen 
by  12. 


§§  576-579] 


MARKING  GOODS 


217 


WRITTEN    EXERCISE 


Find  the  selling  price  per  article: 


Gain  to  be 
realized 


Cost  per 
dozen 


1.  $15.00 

2.  $  9.00     25% 
8.  $27.50     6J% 

4.  $28.00      16|% 

5.  $  4.00 

6.  $37.50 


Cost  per 
dozen 


Gain  to  be 
realized 


20% 


7.  $7.50  18|% 

8.  $25.25  20% 

9.  $17.50  20% 

10.  $18.90  331% 

11.  $1.50  16f% 

12.  $9.60  66}  % 


Cost  per  Gain  to  be 

dozen  realized 

13.  $36.00  25% 

14.  $116.00  121% 

15.  $35.50  20% 

16.  $29.70 

17.  $28.00 

18.  $56.00  28^% 


577.  To  find  the  price  at  which  goods  must  be  marked  to  insure  a 
given  per  cent  of  gain  or  loss,  the  cost  and  discount  series  being  given. 

578.  Example.    A  seal  sack  cost  a  manufacturer  $  240.    At  what 
price  must  it  be  marked  in  order  that  a  discount  series  of  25%  and 
20%  may  be  allowed  and  a  gain  of  33^%  be  realized  ? 

SOLUTION 

Let  100%  represent  the  marked  price. 

A  discount  series  of  25%  and  20%  is  equal  to  a  direct  discount  of  40%. 

100%  -  40  =  60%,  the  amount  to  be  realized. 

$240  =  the  cost  of  the  sack. 

33|,  or  |  of  $240  =  $80,  the  amount  to  be  gained. 

$240  +  $80  =  $320,  the  amount  to  be  realized. 

Therefore,  60%  =$320. 

l%  =  $5.33f 

100%  =  $533.33,  the  marked  price. 

579.  Therefore  the  following  rule : 

Add  the   required  gain  to  or  subtract  the  required  loss 
from  the  cost  and  divide  by  1  minus  the  rate  of  discount. 

WRITTEN  EXERCISE 

1.  What  must  be  the  asking  price  of  a  watch  costing  $  24  in  order  L- 
to  insure  a  gain  of.33J%  and  allow  the  purchaser  a  discount  of  20%? 

2.  After  buying  lace  at  $8  per  piece,  I  so  marked  it  as  to  allow 
discounts  of  25%  and  20%  from  the  marked  price,  and  yet  so  sell  it 
as  to  lose  but  10%  on  my  purchase.     At  what  price  per  piece  was 
the  lace  marked  ? 


• 


218  PERCENTAGE   AND   ITS  APPLICATIONS  (_§  579 

8.  The  cost  of  manufacturing  silk  ties  being  $  36  per  dozen,  how 
much  must  they  be  marked  that  a  gain  of  16f  %  may  be  realized  by 
the  manufacturer  after  allowing  a  discount  of  25%  and  12i%  ? 

4.  If  a  carriage  be  marked  33^%  above  cost,  what  per  cent  of 
discount  can  be  allowed  from  the  marked  price  and  realize  cost  ? 

5.  If  the  list  price  of  an  article  is  40%  advance  on  the  cost, 
what  other  per  cent  of  discount  than  14^  %  must  be  allowed  to  net 

by  the  sale? 


_ 

WRITTEN  REVIEW 

1.  W.  A.  Briggs  &  Co.  bought  of  B.  A.  Altman  &  Son  invoice 
of  silk  hats  at  $100  per  dozen,  less  30%  and  10%.  .  What  price  per 
hat  must  be  asked  in  order  to  gain  33^%  ? 

2.  If  goods  are  retailed  at  an  advance  of  25%,  what  is  the  sell- 
ing price  per  article  of  goods  costing  by  the  dozen  as  follows  :  shoes, 
$48;  boots,  $38.40;  rubbers,  $8.16? 

8.  Briggs,  Slote  &  Co.  imported  hosiery  and  knit  goods  costing 
per  dozen  as  follows:  Hosiery,  $2.40,  $2.50,  $3.84,  and  $4.80; 
knit  goods,  $7.20,  $4.08,  $16.32,  and  $10.40.  Determine  the  sell- 
ing price  per  article,  goods  to  be  sold  at  a  gain  of  25%. 

4.  Using  the  word  handsomely  with  repeater  R  for  the  buying 
key,  and  the  words  black  horse  with  repeater  W  for  the  selling  key, 
mark  the  cost  and  selling  price  for  the  articles  in  problem  3. 

5.  What  should  be  the  marked  price  per  article  of  the  following 
goods  so  as  to  gain  33^%  and  allow  discounts  of  10%  and  10%  ? 
Hats,  per  dozen,  $16.20;  gloves,  per  dozen,  $8.10;  ties,  per  dozen, 
$4.05. 

6.  At  what  price  should  the   following   goods  be  marked  per 
article  so  as  to  allow  discounts  of  25%  and  20%  and  still  net  a  gain 
of  331%  on  the  cost  ?     Hats,  $  9,  $  6.30,  $  15,  $  21  ;  gloves,  $  10.80, 

$12.60,  and  $8.10, 
-    •  !.ir£ 

7.  What  price  each  must  be  asked  for  cocoanuts  costing  $  4  per 

C  that  an  allowance  of  16f  %  for  breakage,  20%  for  decay,  and  11|% 
for  bad  debts  may  be  made,  and  still  a  gain  of  33^%  be  realized  ? 

8.  Having   paid  40^  per   pound  for   tea,  at  what  retail   price 
must  it  be  marked  that  I  may  allow  12|%  for  bad  debts  and  gain 

on  the  cost  ? 


§§  579-58C]  COMMISSION  219 

9.  A  publisher's  prices  are  75%  above  cost.  If  he  allows  his 
agent  a  commission  of  20%  and  sells  at  a  discount  of  10%,  what 
per  cent  of  gain  does  he  make  ? 

10.  What  price  per  pound  must  be  asked  for  coffee  costing 
18^  per  pound,  in  order  that  the  seller  may  deduct  10%  from  the 
asking  price  for  bad  debts,  allow  16f  %  for  loss  in  roasting,  and  still 
gain  20  %  on  the  cost  ? 

COMMISSION 

580.  Commission  is  a  compensation  allowed  by  one  person,  called 
the  principal,  to  another,  called  the  agent,  for  the  transaction  of 
business.     It  is  usually  a  percentage  of  the  money  involved  in  the 
transaction.  < 

Thus,  in  the  purchase  of  goods,  it  is  usually  a  percentage  of  the  prime  cost ; 
in  the  sale  of  goods,  a  percentage  of  the  gross  selling  price  j  in  the  collection  of 
a  debt,  a  percentage  of  the  amount  collected. 

581.  The  person  for,  whom  business  is  transacted  is  called  the 
principal,  and  the  person  authorized  to  transact  business  for  another, 
the  agent,  broker,  commission  merchant,  or  collector,  according  to  the 
nature  of  the  business  transacted. 

582.  A  quantity  of  goods  sent  away  to  be  sold  on  commission  is 
called  a  shipment ;  a  quantity  of  goods  received  to  be  sold  on  com- 
mission is  called  a  consignment.     The  person  who  sends  the  goods  is 
called  the   consignor,  and  the  person  to  whom  they  are  sent,  the 
consignee. 

583.  Guaranty  is  a  percentage  charged  by  an  agent  for 'assuming 
the  risk  of  loss  from  sales  made  by  him  on  credit,  or  for  giving 
pledge  of  the  grade  of  goods  bought. 

584.  Account  sales  is- an  itemized  statement  rendered  by  an  agent 
to  his  principal,  showing  in  detail  the  sales  of  goods  and  charges 
thereon,  together  with  the  net  proceeds  remitted  or  credited. 

585.  Account  purchase  is  an  itemized  statement  rendered  by  a 
purchasing  agent  to  his  principal,  showing  the  quantity,  grade,  and 
price  of  goods  purchased,  and  all  expenses  incurred,  together  with 
the  gross  cost  of  the  transaction. 

586.  The  gross  proceeds  of  a  sale  or  collection  is  the  total  amount 
received  by  an  agent. 


220  PERCENTAGE  AND  ITS   APPLICATIONS         [§§  587-589 

587.  The  net  proceeds  is  the  amount  remaining  after  commission 
and  all  other  charges  have  been  deducted. 

588.  Computations  in  commission  are  performed  in  accordance 
with  the  general  rules  of  percentage,  the  gross  selling  price,  or  the 
prime  cost,  corresponding  to  the  base;  the  per  cent  of  commission, 
either  for  buying  or  selling  or  for  guaranty  of  quality  or  credit,  to 
the  rate;  the  commission  to  the  percentage  ;  the  total  cost  of  the  goods 
bought  by  a  purchasing  agent  to  the  amount;  and  the  proceeds  to  the 
difference. 

DRILL  EXERCISE 

1.  Given  the  amount  and  the  rate,  how  do  you  find  the  base  ? 
percentage  ?  difference  ?  why,  in  each  case  ? 

2.  Compare  the  general  principles  governing  commission  with 
those  governing  abstract  percentage. 

8.  Give  the  five  necessary  formulae  for  performing  the  operations 
in  commission. 

589.  To  find  the  commission,  the  cost  or  selling  price  and  per  cent 
of  commission  being  given. 

DRILL  EXERCISE 

1.  Give   a  short  method  for   finding   10%    commission;   2-J% 
commission. 

SOLUTION.  10%  is  -fa  of  100%,  the  whole  of  a  number.  Hence,  to  com- 
pute commission  at  10%,  find  TV  of  the  base  by  pointing  off  one  place  to  the  left. 

2£%  is  |  of  10  %.  Hence,  to  find  a  commission  of  2|%,  point  off  one  place  to 
the  left  and  divide  by  4. 

2.  Give  a  short  method  for  calculating  a  commission  of  3i%  ; 

;  25%;  331%;  12*96;  6f% 


ORAL  EXERCISE 

By  inspection  find  the  commission  in  the  following  problems  : 

Gross  Selling            Rate  of                                           Prime  Rate  of 

Price                 Commission                                         Cost  Commission 

1.  $945.80              10%                          7.  9  8480.80  25% 

2.  $724.80                5%                          8.  $1200.00  7% 

3.  $440.40                4%                          9.  $1500.00  8% 

4.  $780.20                5%                        10.  $2500.00  14% 

5.  $750.60            33J%                        11.  $2500.00  16% 

6.  $225.00              3£%                        18.  $2978.95  10% 


§§  589-590]  COMMISSION  221 

WRITTEN  EXERCISE 

1.  A  real  estate  agent  sold  a  farm  of  90  acres  at  $  125  per  acre 
on  a  commission  of  2%.     What  was  the  amount  of  his  commission  ? 
How  much  did  he  turn  over  to  his  principal  ?  fyrft}.  £.2^^ 

2.  An  agent  sold  450  barrels  of  flour  at  $  6.25  per  barrel  on  a 
commission  of  3J%.     What  was  his  commission? 

S.  A  collector  succeeded  in  collecting  80%  of  a  doubtful  account 
of  $1500.  If  he  charged  1%%  commission,  how  much  did  he  turn 
over  to  his  principal  ?  &  /  )  )  0 

4.  My  Chicago  agent  buys  for  me  4500  bushels  of  wheat  at  83J^ 
per  bushel.     How  much  should  I  remit  him  to  cover  the  cost  of  the 
wheat  and  his  commission  of  5°/0  ? 

590.  To  find  the  rate  of  commission,  the  commission  and  gross 
sailing  price  or  prime  cost  being  given. 

ORAL  EXERCISE 

Find  the  rate  of  commission  in  each  of  the  following  examples : 

Gross  Selling  Price     Commission  Prime  Cost      Commission 

1.  $750  $7.50  6.  $105  $35 

5.  $216  $6.48  7.  $2400  $60 

3.  $135  $4.04  £  $125  $25 

4.  $150  $5.00  9.  $920  $23 

5.  $2500  $50.00  10.   $200  $24 
11-15.   Find  the  net  proceeds  in  problems  1-5  inclusive. 
16-20.   Find  the  gross  cost  in  problems  6-10  inclusive. 

WRITTEN   EXERCISE 

1.  A  lawyer  collected  a  note  of  $  2500  and  paid  to  his  principal 
$  2437.50.     What  was  his  rate  of  commission  ? 

2.  A  commission  merchant  sold  a  consignment  of  1200  barrels 
of  beef  at  $  14.50  per  barrel.     After  deducting  $  80  for  freight,  $  20 
for  storage,  and  his  commission,  he  remits  his  principal  $  16,952  as 

the  net  proceeds  of  the  sale.     What  was  his  rate  of  commission  ?  f*,r  \  v) 


222  PERCENTAGE   AND   ITS  APPLICATIONS  [§591 

591.  To  find  the  investment  or  gross  sales,  the  commission  and 
per  cent  of  commission  being  given. 

DRILL  EXERCISE 

1.  Formulate  a  short  method  for  finding  the  gross  sales  when 
the  commission  is  given  and  the  rate  is  10%  ;  1%%. 

SOLUTIONS.  10%  equals  ^  of  a  number.  Hence,  if  the  commission  at  10% 
is  given,  the  gross  sales  may  be  found  by  multiplying  the  commission  by  10,  or 
hy  removing  the  decimal  point  one  place  to  the  right. 

7£%  increased  by  ^  of  itself  equals  10%.  Hence,  to  find  the  gross  sales  when 
the  commission  at  7£%  is  given,  remove  the  decimal  point  one  place  to  the  right 
and  add  J. 

2.  Formulate  a  short  method  for  finding  the  gross  sales  when 
the  commission  is  given  and  the  rate  is  2J%  j  3J%  ;  If  %  ;  25%  ; 

;  16f%  ;  ' 


ORAL  EXERCISE 

Find  the  prime  cost  when  :  Find  the  gross  sales  when  : 

1.  5%  commission  =$27.50.  <>$*     7.   7%%  commission  =$90. 

2.  2£%  commission  =$22.50.  8.   16f%  commission  =$150. 

3.  3J%  commission=$  14.20.  [         9.   &|%  commission  ==$11. 

4.  If  %  commission  =$15.50.         10.   25%  commission  =  $140. 

5.  16|%  commission=$  75.50.       11.   1J%  commission=$110. 

6.  71%  com  mission  =$75.75.         12.    16%  commission  =$  640. 
18-18.    Find  the  gross  cost  in  problems  1-6  inclusive. 

19-24-   Find  the  net  proceeds  in  problems  7-12  inclusive. 

WRITTEN  EXERCISE 

1.  What  must  an  agent's  sales  for  one  year  aggregate  in  order 
that  at  «°>%  commission  his  yearly  income  may  be  $2700? 

2.  A  Mobile  factor  earned  $99.75  by  selling  cotton  at  2|%  com- 
mission.    How  many  bales,  averaging  560  pounds,  did  he  sell,  the 
price  being  15  ^  per  pound  ? 


§§  591-593]  COMMISSION  223 


3.  I  paid  a  grain  dealer  1^%  for  buying  corn  for  me  at  62^  per 
bushel.     If  his  commission  amounted  to  $  83.70,  how  many  bushels 
did  he  buy  ? 

4.  An  agent  charged  $433.60  for  selling  a  consignment  of  canned 
fruit.     If  his   rate   of   commission  was   2^%,   what   was   the   net 
proceeds  ? 

592.  To  find  the  investment  and  commission  when  both  are  included 
in  the  remittance  by  the  principal. 

593.  Example.     I   sent  my  agent  $1025  with  instructions  to 
deduct  his  commission  of  21%  and  invest  the  balance  in  wheat. 
How  much  did  he  invest,  and  what  was  his  commission? 

SOLUTION 

Represent  the  actual  investment  by  100%. 

2£%  =  the  charges  for  buying. 

100%  +  2f  %  =  102£  %,  the  cost  of  the  investment  to  the  principal. 

$  1025  =  the  cost  of  the  investment  to  the  principal. 

Therefore,  102£  %  =  $1025. 

1%  =  $10. 

100%  =  $1000,  the  actual  investment  in  wheat. 

$1025  —  $1000  =  $25,  the  commission  for  buying. 


ORAL  EXERCISE 

By  inspection  find  the  amount  to  invest  and  the  commission  in 
each  of  the  following  problems  : 

Rate  of  Rate  of  Rate  of 

Amount  Re-    Commis-  Amount  Re-  Commis-  Amount  Re-  Commis- 

mitted  sion  mitted          sion  mitted  sion 

L   $1030      3%  4.   $315       5%  7.  $515          3% 

2.  $105        5%  5.   $624        1%  8.    $410          2%% 

3.  $550      10%  6.   $205       21%  9.   $2075        3% 

10.  I  sent  my  agent  a  certain  amount  with  which  to  buy  silks, 
after  deducting  his  commission  of  3%.  If  his  commission  was  $30, 
what  was  the  amount  of  my  remittance  ?  /  0  3 


224  PERCENTAGE   AND   ITS  APPLICATIONS  [§593 

WRITTEN  SXERCISE 

L  How  many  pounds  of  wool,  at  27  ^  per  pound,  can  be  bought 
for  $8424,  if  the  agent  is  allowed  4%  for  purchasing? 

#.  I  remitted  $1306.45  to  a  Boston  agent  for  the  purchase  of 
soft  hats.  If  the  agent's  commission  is  4%,  and  he  makes  an  added 
charge  of  2%  for  guaranty  of  quality,  how  many  dozen  hats,  at 
$8.50  per  dozen,  should  he  send  me  ?  \  (L  ^ 

8.  A  city  merchant  remitted  his  country  agent  Jfl>  1093.60  with 
which  to  buy  butter.  If  the  agent's  charges  were  3%  commission, 
5%  guaranty,  ana  $13.60  for  inspection,  how  many  pounds,  at  25  £ 
per  pound,  did  he  buy,  and  what  was  his  commission  ? 

4-  I  remitted  $300  to  an  agent  for  the  purchase  of  hops.  If 
the  agent's  charges  were  5%  for  purchase  and  $6  for  inspection, 
how  many  pounds  at  16^  per  pound  ought  he  to  buy  ?  ^  j[)j 

nil 

ORAL  REVIEW 

1.  If  you  sell  books  to  the  amount  of  $240  on  33|%  commis- 
sion, what  amount  do  you  earn,  and  what  is  the  net  proceeds  of  the 
sale? 

2.  I  send  you  $  205  with  instructions  to  expend  it  for  wheat  at 
$  1  per  bushel,  after  retaining  your  commission  of  2|%.     How  many 
bushels  of  wheat  will  you  be  able  to  buy,  and  what  will  be  your 
commission  ? 

8.  If  I  -remit  $95  as  the  proceeds  of  an  account  collected  by 
me,  how  much  have  I  retained,  my  rate  of  commission  being  5%  ? 

4-  What  is  the  amount  of  sales  when  the  net  proceeds  are  $  975 
and  the  commission  21%  ? 

5.  An  agent  charges  5%  commission  and  receives  $250.     Find 
the  net  proceeds  of  the  consignment. 

6.  A  manufacturer  sent  his  purchasing  agent  $  510  with  which 
to  buy  leather,  after  deducting  his  commission.    If  the  agent  received 
$  10  for  his  services,  what  was  his  rate  of  commission  ? 

WRITTEN  REVIEW 

1.  Eule  a  sheet  of  paper,  copy  the  following  account  sales,  | 
and  make  the  necessary  extensions,  etc. : 


693] 


COMMISSION 


ACCOUNT  SALES 

BOSTON,  MASS.,  Feb.  23,  1904. 
Sold  for  the  Account  of 

E.  W.  HARDEN,  Worcester,  Mass. 
BY  E.  A.  REED  &  Co.,  COMMISSION  MERCHANTS. 


1904 

Jan. 

25 

100  bbl.  S.  P.  Flour                            6.75 

28 

150bbl.  R.  P.  Floor                           6.60 

Feb. 

18 

200  bbl.  S.  P.  Flour                            6.80 

20 

100  bbl.  S.  P.  Flour                            6.00 

CHARGES 

Jan. 

15 

Freight,  $135                 Cartage,         $26 

Feb. 

20 

Storage,  $16.50              Insurance,  $6.20 

23 

Guaranty,  1%                Commission,  5% 

Net  proceeds, 

2.  Prepare  an  account  sales  under  date  of  Feb.  24  for  5000  bu. 
of  wheat,  sold  by  E.  L.  Hardy  &  Co.,  Boston,  Mass.,  for  the  account 
of  Welsh  Bros.  &  Co.,  Springfield,  Mass.  Sales :  Feb.  1,  500  bu.  at 
$1.02;  Feb.  15,  1000  bu.  at  $1.08;  Feb.  19,  500  bu.  at  $1.05;  Feb. 
22,  the  remainder  at  $1.  Charges:  freight,  $95;  cartage,  $18; 
storage,  $17  50;  insurance,  \%  5  guaranty,  1%  ;  commission,  2%. 

8.  Rule  a  sheet  of  paper  and  copy  the  following  account  pur- 
chase, making  the  necessary  extensions,  etc. 

ACCOUNT  PUBCHASE 

BOSTON,  MASS.,  Feb.  28,  1904. 
Purchased  by  F.  B.  BEBBIMAN  &  Co., 

For  the  Account  and  Risk  of 

E.  L.  BROWN,  Paterson,  N.J. 


3 
4 
5 
8 

half-ch.  G.  Tea,  165  Ib.                     34^ 
half-ch.  O.  Tea,  240  Ib.                      41  j* 
half-ch.  J.  Tea,  350  Ib.                       23^ 
mats  J.  Coffee,  600  Ib.                       24  ^ 

CHARGES 
Cartage                                           $7.90 
Commission,  2  % 

Amount  charged  to  your  account, 

226  PERCENTAGE  AND  ITS  APPLICATIONS  [§590 

4-  In  accordance  with  the  foregoing  form  prepare  an  account 
purchase  of  tea  purchased  by  W.  L.  Jordan  &  Co.,  Feb.  23,  for  the 
account  and  risk  of  Adams,  Kand  &  Co.  Purchases  :  10  half-chests 
J.  Tea,  600  lb.,  at  38^;  5  half-chests  0.  Tea,  250  lb.,  55^;  5  cases  C. 
Tea,  250  lb.,  at  55^;  8  half-chests  E.  B.  Tea,  480  lb.,  45^.  Charges  : 
cartage,  $7.50;  commission,  2%. 

s  5.  I  place  a  claim  of  $2580  in  the  hands  of  an  attorney  for 
collection.  If  the  debtor  is  a  bankrupt  having  liabilities  aggregating 
$18,000  and  resources  aggregating  $13,500,  how  much  should  I 
receive  after  my  attorney  has  deducted  his  commission  of  2  %  ? 

x  6.  A  collector  obtained  75%  of  the  amount  of  an  account,  and 
after  deducting  12%  for  fees  remitted  his  principal  $495.  What 
was  the  amount  of  his  commission  ? 

7.  A  Hartford  fruit  dealer  sent  a  Lockport  agent  $  1946.70,  and 
/-instructed  him  to  buy  apples  at  $1.40  per  barrel.     The  agent  charged 

/  3%  for  buying,  and  shipped  the  purchase  to  his  principal  in  six  car 
loads  of  an  equal  number  of  barrels.  How  many  barrels  did  each 
car  contain  ? 

8.  Find  the  per  cent  of  commission  on  a  purchase  if  the  gross 
cost  is  $2048.51,  the  commission  $87.30,  the  cartage  $20,  and  other 
charges  $1.21.    ' 


9.  A  collector  obtained  75%  of  a  doubtful  account  of  $1750. 
How  much  was  his  per  cent  of  commission  if,  by  agreement  with 
the  principal,  the  commission  was  to  be  50%  of  the  net  proceeds 
remitted?  v./:  '••, 

10.  A  farmer  received  from  his  city  agent  $  490  as  the*  net  pro- 
ceeds of  a  shipment  of  butter,.     If  the  agent's  commission  is  3%, 
delivery  charges  $  6.80,  and  5  %  charge  is  made  for  guaranty  of  quality 
to  purchasers,  how  many  pounds,  at  27^  per  pound,  must  have  been 
sold,  and  how  much  commission  was  allowed  ?  A  <><>^  %*  /7  /*  •«*  o 

11.  An  agent  sold  2000  bu.  Alsike  clover  seed  at  $7.85  per 
bushel,  on  a  commission  of  5%,  and  1200  bu.  medium  red  at  $5.20, 
on  a  commission  of  2|%,  taking  the  purchaser's  3-month  s'  note  for 
the  amount  of  the  sales.     If  the  agent  charges  4%  for  his  guaranty 
of  the  note,  what  amount  does  he  earn  by  the  transaction  ? 


§  593]  COMMISSION  227 

12.  Find  the  net  proceeds  of  a  sale  made  by  an  agent  charging 
if  incidental  charges  and  commission  charges  were  each  $41.30. 

13.  Find  the  gross  proceeds  of  a  sale  made  by  an  agent  charging 
for  commission,  5%  for  guaranty,  $17.65  for  cartage,  $11.40 

for  storage,  and  $3.25  for  insurance,  if  the  net  proceeds  remitted 
amount  to  $  1714.10. 

14.  I  sent  $3402.77  to  my  Atlanta  agent  for  the  purchase  of 
sweet  potatoes  at  $  1.60  per  barrel ;  his  charges  were,  for  commission, 
2J%;   guaranty,  3%;    dray  age,  1^  per  barrel;   and  freight,  $200. 
How  many  barrels  did  he  buy,  and  how  much  unexpended  money 
was  left  in  his  hands  to  my  credit  ?  ^^  ^/  // 

15.  I  received  from  Duluth  a  cargo  of  16,000  bu.  of  wheat,  which 
I  sold  at  $1.10  per  bushel,  on  a  commission  of  4%;  by  the  con- 
signor's instructions  I  invested  the  net  proceeds  in  a  hardware  stock, 
for  which  I  charged  5%  commission.     What  was  the  total  commis- 
sion, and  how  much  was  invested  in  hardware  ? 

16.  Having  sent  a  New  Orleans  agent  $1835.46  to  be  invested 
in  sugar,  after  allowing  3%  on  the  investment  for  his  commission 
I  received  32,400  pounds  of  sugar.     What  price  per  pound  did  the 
sugar  cost  the  agent  ? 

17.  An  agent  in  Providence,  R.I.,  received  $828  to  invest  in 
prints,  after  deducting  his  commission  of  3|%.     If  he  paid  7^  per 
yard  for  ths  prints,  how  many  yards  did  he  buy  ? 

18.  An  agent  sold,  on  commission,  1750  barrels  of  mess  pork  at 
$  16.50  per  barrel,  and  508  barrels  of  short  ribs  at  $  18  per  barrel, 
charging  $  112.50  for  cartage  and  $  5.55  for  advertising.     He  then 
remitted  to  his  principal  $36,000,  the  net  proceeds.     Find  the  rate 
of  commission,     tv 

19.  Render  in  full  the  following  account  sales,  supplying  rates 
per  cent  for  insurance  and  commission,  and  showing  net  proceeds : 
Feb.  23,  Emery  Williams  &  Co.,  Troy,  N.Y.,  sold  for  Moody  Bros. 
&  Co.,  Eome,  K  Y.,  12,000  lb.-  wool  at  35  £  12,000  yd.  woolen  goods  at 
75  f.     Charges  :  freight,  $  450 ;  insurance,  $  33 ;  commission,  $  264. 

20.  An  agent  sold  wheat  on  5%  commission  and  invested  the 
proceeds  in  barley  at  75^  per  bushel  on  a  commission  of  5f%.     If 
his  total  commission  was  $1200,  how  many  bushels  of  barley  did 
he  buy  ? 


228  PERCENTAGE  AND   ITS  APPLICATIONS         [§§694-601 

INTEREST 

594.  Interest  is  that  which  is  paid  for  the  use  of  money. 

595.  The  essential  elements  of  interest  are  the  principal,  the  time, 
the  rate,  the  interest,  and  the  amount. 

596.  The  sum  upon  which  interest  is  charged  is  termed  the 
principal ;  the  period  for  which  the  principal  bears  interest,  the  time ; 
the  annual  rate  charged  for  the  use  of  the  principal,  the  rate  of 
interest ;  the  product  of  the  rate  of  interest  and  the  time,  the  per 
cent  of  interest ;  the  result  obtained  by  taking  a  per  cent  of  interest 
of  the  principal,  the  interest ;  the  sum  of  the  principal  and  interest, 
the  amount. 

597.  Legal  interest  is  interest  computed  at  the  rate  established 
by  law  to  apply  when  no  agreement  is  made.      The  legal  rate  of 
interest,  being  established  by  state  statutes,  varies  in  the  different 
states. 

598.  Usury  is  any  rate  of  interest  in  excess  of  the  legal  rate. 

In  a  number  of  the  states  parties  may,  by  special  agreement,  receive  interest 
at  a  higher  rate  than  the  legal  rate. 

A  person  taking  a  usurious  rate  of  interest  is  liable  to  certain  penalties 
regulated  by  state  statutes. 

SIMPLE  INTEREST 

599.  Simple  interest  is  interest  allowed  for  the  use  of  the  princi- 
pal only. 

600.  The  term  interest  is  always  understood  to  mean  simple 
interest.     If  other  forms  of  interest  are  meant  they  are  specifically 
designated ;  as,  compound  interest,  periodic  interest. 

601.  For  convenience,  interest  is  usually  computed  on  the  basis 
of  the  commercial  year  of  12  months  of  30  days  each,  or  360  days. 
Interest  on  this  basis  is  called  common  interest.      The  practice  of 
taking  360  days  as  a  year,  being  sufficiently  exact  for  business  pur- 
poses, has  the  sanction  of  law  in  some  states  and  is  generally  used 
in  all  the  states. 


§§  G02-604]  INTEREST 

602.  Simple  interest  is  an  application  of  the  principles  of  abstract 
percentage  with  the  additional  element  time  introduced.     The  prin- 
cipal in  interest  corresponds  to  the  base  in  percentage ;  the  per  cent 
of  interest,  to  the  rate ;  the  interest,  to  the  percentage ;  and  the  sum 
of  the  principal  and  interest,  to  the  amount.     The  solution  of  problems 
in  interest  is  therefore  dependent  upon  the  general  principles  of 
abstract  percentage, 

603.  There  are  many  methods  of  computing  simple  interest,  but 
those  given  herewith  are  the  most  rational  and  simple.    The  ordinary- 
day  and  the  bankers'  sixty-day  methods  are  particularly  adapted  to 
finding  the  interest  when  the  time  is  expressed  in  days,  and  the  six 
per  cent  method  to  finding  the  interest  when  the  time  is  expressed  in 
years  and  months,  or  years,  months,  and  days. 


Ordinary-day  Method 
DRILL  EXERCISE 

1.  What  is  the  interest  on  $  1  for  1  year  at 

2.  What  part  of  a  commercial  year  is  60  days  ?  6  days  ? 

8.  How  many  days  will  it  take  $1  to  yield  1  cent  interest? 
1  mill  interest  ? 

4.  What  is  the  interest  on  $1  for  60  days  at  6%  ?  for  6  days  ? 

5.  What  is  the  interest  on  $1  for  1  day  at  6%  ? 

6.  What  is  the  interest  on  $6  for  1  day  at  6%  ?  on  $18?  on 
$36?  on  $300?  on  $1200? 

7.  What  part  of  the  principal  is  the  interest  for  6  days  at  6%  ? 

8.  Give  a  simple  way  to  find  the  interest  on  any  principal  for 
any  number  of  days  at  6%. 

604.  General  Principles.  1.  In  6  days  at  6%  any  principal  will 
yield  interest  equal  to  .001  of  itself ;  in  1  day,  interest  equal  to  .OOOJ 
of  itself. 

2.  .001  of  any  given  principal  is  equal  to  6  times  the  interest  for 
1  day  at  6%. 


230  PERCENTAGE   AND   ITS   APPLICATIONS         [§§  005-606 

605.  Examples.     1.   Find  the  interest  on  $750  for  11  da.  at  6%. 

SOLUTION.     .001  of  the  principal,  or 
.750  X  11  =8.250.  ^  -75?  is  equal  to  c  times  tke  interest 

8.250-5-6  =  1.375,  or  $1.38.        for  1  day.     Hence,   11  times  $.75,  or 

$8.25,  is  equal  to  6  times  the  interest 

for  11  days.     If  6  times  the  interest  for  11  days  is  $8.25,  the  actual  interest  for 
11  days  must  be  £  of  $8.25,  or  $1.38. 

2,   Find  the  interest  on  $875  for  24  da.  at  6%. 

.875  X  4  =  3.500,  or  $  3.50.  SOLUTION.     .001  of  the  principal,  or 

$.875,  is  equal  to  the  interest  for  6  times 

1  day,  or  6  days.     Since  24  days  are  4  times  6  days,  4  times  $  .875  must  be  the 
interest  for  24  days.     4  times  $  .875  =  $  3.50,  the  required  interest. 

606.  Hence,  the  following  rule  may  be  derived: 

Point  off  three  integral  places  in  the  principal,  multiply 
~by  the  number  of  days,  and  divide  by  6.  The  result  is  the 
required  interest  at  6%.  Or, 

When  it  is  seen  that  the  time  in  days  and  months  is  a 
multiple  of  6,  point  off  three  integral  places  in  the  principal 
and  multiply  by  j  of  the  number  of  days.  The  result  is  the 
required  interest  at  6%. 

WRITTEN  EXERCISE 

s. 

At  6%  per  annum  find  the  interest  on^  > 

1.  $  750  for  73  da.  7.  $  476.87  for  95  da.  IS.  $  728.16  for  84  da. 

2.  $840  for  19  da.  8.  $  925.14  for  72  da.  14.  $846.92  for  108  da. 

3.  $  920  for  24  da.  9.  $  724.18  for  75  da.  15.  $  1246.45  for  24  da. 

4.  $  780  for  36  da.  10.  $  420.10  for  11  da.  16.  $  1432.1 8  for  36  da. 

5.  $  920  for  42  da.  11.  $  500.60  for  7  da.  17.  $  1945.62  for  18  da. 

6.  $  924  for  17  da.  12.  $  702.45  for  17  da.  18.  $  7514.95  for  12  da. 

Using  the  exact  number  of  days,  find  the  interest  at  6%  on : 

19.  $  170  from  July  15  to  Sept.  1 ;  from  Apr.  6  to  Oct.  9. 

20.  $  1750  from  Jan.  1  to  Feb.  8 ;  from  May  15  to  July  9. 

21.  $  2470  from  Apr.  7  to  July  1 ;  from  Apr.  2  to  Aug.  1. 

22.  $7562  from  July  2  to  Sept.  5 ;  from  Mar.  2  to  Apr.  30. 
28.   $2172  from  Jan.  2  to  July  9;  from  Sept.  2  to  Dec.  1. 

24.  $2400  from  Oct.  1  to  Dec.  1 ;  from  May  8  to  Aug.  1. 

25.  $  2(575  from  Oct.  5  to  Nov.  1 ;  from  Sept.  9.  to  Dec.  1. 


§§  607-611]  INTEREST  231 

607.  To  find  the  interest  for  any  number   of   days  at  any  rate 
per  cent  per  annum. 

608.  General  Principle.     .001  of  any  given  principal  is  the  interest 
for  1  da.  at  36%. 

609.  Example.     What  is  the  interest  on  $750  for  16  da.  (a)  at 
6%  ?    (6)  at  4%  ?   (c)  at  9%  ?   (d)  at 


SOLUTIONS,     (a)  .001  of  $750  =  $.75,  or 

(a)  .750  X  16  =  12.  the  interest  for  6  days.     $.75  x  16  =  $  12,  or 

12  -f-  6  =  2,  or  $  2.        the  interest  for  6  times  16  days.     $  12  -=-  6  =  $  2, 
or  the  required  interest. 

(6)  .001   of  $750  =  $.75,  the  in- 

:  L*-  terest  for  6  days  at  6%,  or  for  1  day 

12  -h  9  =  1.33,  or  $1.33.  at  36%.     $.75  x  16  =  $12,  or  the  in- 

terest  for  16  days  at  36%.    4%  is  £ 
of  36  %.    Hence,  $  of  $  12,  or  $  1.33,  is  the  interest  at  4  %. 

(c]  750  X  16  =  12  (°)  Tne  interest  on  tne  principal  for  16 

days  at  36  %  is  $  12.     9  %  is  \  of  36  %.     Hence, 
=  3,  O    *3.        ^  of  3  12)  or  $  3,  is  the  interest  at  9  %. 

(d)  The  interest  on  the  principal 

(d)  .750  X  16  =  for  16  days  at  360/o  is  $12.    4jo/fl  is  | 

12  --  8  =  1.50,  or  9  1.50.        of  36%.     Hence,  \  of  $12,  or  $1.50,  is 

the  interest  at  4^%. 

610.  Hence  the  following  rule  may  be  derived  : 

Point  off  three  integral  places  in  the  principal,  multiply 
by  the  number  of  days,  and  take  such  a  part  of  the  product 
as  the  given  rate  per  cent  is  of  36%. 

611.  Deducing  a  rule  for  each  of  the  ordinary  rates  of  interest, 
we  have  the  following  : 

Point  off  three  integral  places  in  the  principal,  multiply 
by  the  number  of  days,  and  to  find  the  interest  at  6%,  divide 
by  6;  at  3%,  divide  by  12;  at  4%,  divide  by  9;  at  41%,  di- 
vide by  8  ;  at  5%,  divide  by  7.2  (8  and  .9)  ;  at  7%,  divide  by  6  and 
add  \  of  the  quotient;  at  7J%,  divide  by  6  and  add  \  of  the 
quotient;  at  8%,  dirhlc  by  4.5  (9  and  .5)  ;  at  9%,  divide  by  4'>  ati 
10%,  divide  by  3.6  (6  and  .6) 


232 


PERCENTAGE   AND   ITS   APPLICATIONS 


[§611 


WRITTEN  EXERCISE 


Find  the 

amount. 

Principal 

Time 

Bate 

1.   $9000 

91  da. 

4% 

8.    $1700 

73  da. 

4% 

S.   $2750 

81  da. 

±% 

4.   $2400 

15  da. 

8% 

5.   $1750 

21  da. 

8% 

6.   $4200 

84  da. 

10% 

7.   $2972.50 

31  da. 

9% 

8.   $1750.90 

87  da. 

9% 

Principal  Tim*  Bate 

9.  $2431.75  35  da.      9% 

10.  $1862.15  34  da.      9% 

/f.  $2417.50  41  da. 

18.  $7500.75  16  da. 

18.  $2400  19  da. 

/£  $1840.75  71  da. 

15.  $2417.92  76  da. 


16.   $1695.14    93  da. 


3% 


Bankers'  Sixty-day  Method 
DRILL  EXERCISE 

1.  How  many  months  will  it  take  $  1  to  yield  1  cent  interest  at 
6%  ?     How  many  days  ? 

2.  What  is  the  interest  on  $75  for  60  days  at  6%  ?  on  $250  ? 
on  $920?  on  $780?  on  $240.75?  on  $21729.75? 

8.   What  part  of  a  number  is  the  interest  at  6%  for  60  days  ? 
^.   What  is  the  simplest  way  to  find  the  interest  on  any  princi- 
pal for  60  days  ? 

5.  What  part  of  60  days  are  30  days  ?    20  days  ?    15  days  ? 
10  days  ? 

6.  What  is  the  interest  on  $84  for  30  days  ?  for  20  days  ?  for 
15  days  ?  for  10  days  ? 

7.  What  is  the  simplest  way  to  find  the  interest  on  any  princi- 
pal for  30  days  at  6%?  for  20  days  ?  for  15  days  ?  for  10  days  ? 

8.  How  long  will  it  take  $1  at  6%  to  yield  1  mill  interest  ? 

9.  What  part  of  the  principal  is  the  interest  at  6%  for  6  days  ? 

10.  What  is  the  interest  on  $175  for  6  days  at  6%  ?  on  $215? 
on  $240?  on  $378?  on  $7560?  on  $8925.75?  on  $4928.79? 

11.  What  is  the  simplest  way  to  find  the  interest  on  any  princi- 
pal for  6  days  at  6%? 


§§  611-612]  INTEREST  233 

12.  What  is  the  interest  on  $  240  for  3  days  ?  2  days  ?  1  day  ? 

13.  How  many  months  at  6%  will  it  take  $  1  to  yield  10  cents 
interest  ?     How  many  days  ? 

14.  What  part  of  the  principal  is  the  interest  at  6%  for  600 
days? 

15.  What  is  the  interest  on  $  800  at  6%  for  600  days  ?  on  $  9500  ? 
on  $2465?  on  $5619?  on  $4500?  on  $217.40?  on  $924.68?  on 
$275.19? 

16.  What  is  the  simplest  way  to  find  the  interest  on  any  princi- 
pal at  6%  for  600  days  ?  300  days  ?  200  days  ?  150  days  ?  75  days  ? 
120  days  ?  100  days  ?  50  days  ? 

17.  In  how  many  days  will  the  interest  at  6%  equal  the  prin- 
cipal ? 

18.  What  is  the  interest  on  $1  for  6000  days  at  6%  ?  on  $24  ? 
on  $97?  on  $55?  on  $372.50?  on  $920.75?  on  $860.44? 

19.  Pointing  off  3  integral  places  in  the  principal  gives  the  inter- 
est for  how  many  days  at  6%  ?  2  places  ?  1  place  ?  taking  the  prin- 
cipal for  the  interest  ? 

20.  What  is  the  interest  on  $5695  for  6  days  at  6%?  for  60 
days  ?  for  600  days  ?  for  6000  days  ? 

612.   General  Principles.     1.   Pointing  off  3  integral  places  from 
the  right  in  the  principal  gives  the  interest  at  6%  for  6  days. 

2.  Pointing  off  2  integral  places  from  the  right  in  the  principal 
gives  the  interest  at  6%  for  60  days. 

3.  Pointing  off  1  integral  place  from  the  right  in  the  principal 
gives  the  interest  at  6%  for  600  days. 

4.  Writing  the  principal  for  the  interest  gives  the  interest  at  6% 
for  6000  days. 

Thus,  the  interest  on  $7621  for  6  days  at  6%  is  $7.52  ;  for  60  days,  $75.21 ; 
for  600  days,  $762.10 ;  for  6000  days,  $7621. 

ORAL   EXERCISE 

Find  the  interest  at  6%  on: 

1.  $360  for  6  days ;  for  3  days ;  for  2  days ;  for  1  day. 

2.  $900  for  60  days;  for  30  days;   for  20  days;   for  15  days; 
for  12  days;  for  10  days. 


234  PERCENTAGE   AND  ITS   APPLICATIONS         [§§  612-615 

8.  $100  for  18  days;  for  24  days;  for  36  days;  for  42  days; 
for  48  days  ;  for  54  days  ;  for  66  days. 

4.  $200  for  180  days;  for  240  days;  for  420  days;  for  480 
days  ;  for  546  days  ;  for  660  days. 

6.   $240  for  7  days. 

SOLUTION.  7  days  are  \\  times  6  days.  Hence,  to  find  the  interest  for  7 
days,  point  off  3  places  from  the  right  in  the  principal-  and  add  g.  $  .24  -f  J  of 
itself  =  $.28,  the  required  interest. 

6.  $360  for  7  days  ;  for  8  days  ;  for  9  days. 

7.  $990  for  4  days. 

SOLUTION.  4  days  are  £  less  than  6  days.  Hence,  to  find  the  Interest  for  4 
days,  point  off  3  places  from  the  right  in  the  principal  and  subtract  |.  $.99  —  \ 
of  itself  —  $.66,  the  required  interest. 

8.  $240  for  4  days;  for  5  days;  for  40  days;  for  50  days;   for 
40  days  ;  for  80  days  ;  for  90  days  ;  for  70  days  ;  for  50  days. 

613.  To  find  the  interest  at  6%  for  aliquot  parts  of  6  or  60  days,  or 
aliquot  parts  more  or  less  than  6  or  60  days. 

614.  Examples.    1.   What  is  the  interest  on  $  1240  for  30  days 
at6%? 

$  12  40  SOLUTION.     .01  of  the  principal,  or  $12.40  is  the  interest  for 

'oA     6°  days'    30  days  is  $  of  60  days.     Since  $12.40  is  the  interest 
e  interest  for  3Q  days  is  £  Of  $  12.40,  or  $6.20. 


&  What  is  the  interest  on  $2400.60  for  80  days  at  6  %  ? 
$  24  0060  SOLUTION.     .01  of  the  principal,  or  $  24.006,  is 

8  0020  ^e  *nterest  *or  ^°  days.    80  days  are  $  more 


than  60  days.     Hence,  4  more  than  $24.006,  or 
$32.0080,  or  $32.01.    $3201j  is  the  required  in\erest 

8.  What  is  the  interest  on  $360  for  5  days  at  6%  ? 

$  .360  SOLUTION.     .001  of  the  principal,  or  $  .36  is  the  interest  for  6 

.060       days.    Since  5  days  are  J  less  than  6  days,  the  interest  for  5  days 
<c  OA          is  4  less  than  $.36,  or  $ .80. 

*JP  .U\J 

615.     Therefore  the  following  rule  may  be  derived : 

For  6  days  point  off  three  integral  places  from  the  right 
in  the  principal,  and  for  60  days,  point  off  2  integral  places. 

Take  such  a  part  of  the  interest  for  6  or  60  days  as  the 
given  number  of  days  are  a  part  of  6  or  60  days.  Or, 


§615]  INTEREST  235 

Take  such  a  part  of  the  interest  for  6  or  60  days  as  the 
given  number  of  days  is  a  part  more  or  less  than  6  or  60 
days. 

DRILL   EXERCISE 

1.  What  aliquot  part  less  than  60  days  are  55  days  ?  50  days  ? 
40  days  ?   45  days  ? 

2.  What  aliquot  part  more  than  60  days  are  65  days  ?  70  days  ? 
75  days  ?  80  days  ?  90  days  ? 

8.  What  aliquot  part  more  than  6  days  are  7  days?  8  days? 
9  days  ? 

4.  What  aliquot  part  less  than  6  days  are  5  days  ?  4  days  ? 

5.  Give  a  simple  way  to  find  the  interest  at  6%  for  80  days; 
for  90  days ;  for  70  days ;  for  50  days ;  for  45  days ;  for  40  days ; 
for  5  days ;  for  7  days ;  for  4  days ;  for  8  days ;  for  9  days. 

WRITTEN  EXERCISE 

1.  Find  the  total  amount  of  interest  at  6%  on: 

9  2400  for  60  da.  9  440  for  15  da.  $  720  for  2  da.  9  840  for  1  da. 
$  1200  for  30  da.  $  555  for  12  da.  $  240  for  3  da.  $  640  for  15  da 
$900  for  20  da.  $660  for  10  da.  $840  for  6  da.  $810  for  20  da. 

8.  Find  the  total  amount  of  interest  at  6%  on: 

$  450  for  20  da.  $  680  for  45  da.  $  990  for  80  da.  $  660  for  5  da. 
$  720  for  50  da,  $  820  for  75  da.  $  370  for  90  da.  $  750  for  7  da. 
$  810  for  40  da.  $  960  for  70  da.  $  740  for  3  da.  $  930  for  4  da. 

8.   Find  the  total  amount  of  interest  at  6%  on: 

$  1152  for  8  da.  $  1700  for  7  da.  $  439.17  for  50  da, 

$  1600  for  20  da.  $  2100  for  90  da.  $  3100  for  40  da. 

$  519  for  15  da.  $  975.49  for  70  da.  $  1350.90  for  10  da. 

$  2150.42  for  50  da.  $  832.65  for  90  da,  $  759.18  for  1  da, 

4-  Find  the  total  amount  of  interest  at  6%  on : 

$  1800  for  65  da.  $  7421.18  for  6  da.  $  640  for  150  da. 

$  1200  for  55  da.  $  7246  for  40  da.  $  1260  for  300  da. 

$  9128.77  for  7  da.  $  8400  for  45  da.  $  799.49  for  600  da 

$  3160.90  for  2  da.  $  750  for  200  da.  $  9600  for  5  da. 


236  PERCENTAGE   AND    ITS   APPLICATIONS         [§§616-018 

616.  To  find   the  interest   at  6%    when  the  days   are   an  even 
number  of  times  6  or  60. 

617.  Examples.     1,  Find  the  interest  on  $  690  for  180  da.  at  6  %  - 

$690  SOLUTION.     180  days  are  3  times  60  days.     .01  of  $690,  or 

'  $6.90,  is  the  interest  for  60  days.     Since  180  days  are  3  times 

$  20.70        60  dayg^  3  timeg  $6<90i  or  $20.70,  is  the  required  interest. 


2.   Find  the  interest  on  $2100  for  54  da.  at  6$. 

SOLUTION.     54  days  are  9  times  6  days.     .001  of  $2100,  or 
$2.100        $2.10,  equals  the  interest  for  6  days.     Since  54  days  are  9 
$  18.90          times  6  days,  the  interest  for  54  days  will  be  9  times  $2.10,  or 
$18.90. 

618.   Therefore  the  following  rule  : 

Find  the  interest  for  6  or  60  days,  and  multiply  the  result 
by  the  number  of  times  that  6  or  60  days  is  contained  in 
the  given  number  of  days. 

WRITTEN  EXERCISE 

1.  Find  the  total  amount  of  interest  at  6%  on: 

$  925  for  54  da:  $  350  for  18  da.  $  4100  for  360  da, 

f  340  for  36  da.  $  311  for  66  da.  $  917  for  420  da. 

$  420  for  40  da.  $  710  for  24  da.  $  700  for  42  da. 

$  19  for  180  da.  $  3100  for  240  da.  $  419.20  for  18  da. 

2.  Find  the  total  antount  of  interest  at  6%  on: 

$  755  for  180  da.  ^    '   $  3100  for  54  da.  $  179.11  for  JL80  da.   l 

$  101.18  for  54  da.  ^   ^  1700  for  36  da.  $  &0.18  for  240  da.  \  » 

$500.11  for^66  da.  s  >  b   $  1100.59  for  48  da.  f  710.18  for  420  da. 

$  2100  for  42  da.  $  317.42  fgrjjffl  da.  '  $  111.49  for  18  da. 

S.   Find  the  total  amount  of  interest  at  6%  on: 

$  519  for  24  da.  $  1900  for  36  da.  $  1100  for  18  da. 

$  1600.53  for  54  da.  $  170.50  for  240  da.  $  1700  for  120  da. 

f  11  for  540  da.  $  214.18  for  18  da.  $  210.40  for  66  da 

$  210.90  for  180  da.  $  167.90  for  540  da.  $  1100  for  72  da 


§§  618-621]  INTEREST  237 

4.  Find  the  total  amount  of  interest  at  6%  on : 

$  121  for  18  da.  $  760  for  240  da.  $  900  for  30  da. 

$  745  for  600  da.  $  500  for  42  da.  $  800  for  60  da. 

$  600  for  120  da.  $  360  for  72  da.  $  788  for  66  da. 

$ 20  for  36  da.  $  350  for  180  da.  $  89  for  54  da. 

619.  To  find  the  interest  at  6%  for  any  number  of  days. 

620.  Examples.     1.   What  is  the  interest  on  $660  for  11  days 

at6%? 

SOLUTION.     Sometimes  aliquot  parts  may  be  subdivided  so  as 
$6.60       to  make  two  or  more  aliquot  parts.    Subdividing  11  we  have  10 
and  1,  or  \  of  60  and  \  of  6.    The  interest  for  60  days  is  $6.60, 
and  for  10  days  $  1.10.     The  interest  for  6  days  is  $.66,  and  for 
$  1  21       1  day  $  -11-     Adding  the  interest  for  10  days  and  the  interest  for 
1  day,  the  required  interest  is  found  to  be  $  1.21. 

2.   What  is  the  interest  on  $240  for  53  days  at  6%  ? 

SOLUTION.     53  days  are  1  day  less  than  9  times  6  days.     The 
$.240       interest  for  6  days  is  $  .24,  and  for  54  days  $  2.16.     If  the  interest 
2.160       for  6  days  is  $  .24,  the  interest  for  1  day  is  $.04.     If  the  interest 
.04         for  54  days  is  $2.16,  and  the  interest  for  1  day  is  $.04,  the  in- 
$2  12         terest  for  53  days  is  the  difference  between  $2.16  and  $.04,  or 
$2.12. 

5.  What  is  the  interest  on  $240  for  127  days  at  6%  ? 

SOLUTION.  127  =  60  x  2  +  6  +  1.  The  interest  for  60  days  is 
$  2.40.  Hence  the  interest  for  120  days  is  twice  $  2.40,  or  $4.80. 
The  interest  for  6  days  is  $.24,  and  the  interest  for  1  day  is  $  of 
$  .24,  or  $  .04.  Adding  the  interest  for  120  days,  6  days,  and  1  day 
we  have  $  5.08,  or  the  interest  for  127  days. 

621.  Therefore  the  following  rule : 

Find  the  interest  on  the  principal  for  6  days  by  pointing 
off  three  integral  places  from  the  right  in  the  principal,  and 
for  60  days  by  pointing  off  two  integral  places. 

For  any  number  of  days  take  such  a  part  of  the  interest 
for  6  days,  or  for  60  days,  as  the  given  number  of  days  is  a 
part  more  or  less  than  6  days,  or  60  days;  or  as  many  times 
the  interest  for  6  days,  or  60  days,  as  the  required  number  of 
days  will  contain  6  or  60  any  multiple  of  6  or  60  days. 


238 


PERCENTAGE   AND  ITS   APPLICATIONS         [§§621-624 


WRITTEN   EXERCISE 

Principal  Time 

13.  $325.50  29  da. 

14.  $211.10  57  da. 
16.  $440  25  da. 

16.  $  309.09  83  da. 

17.  $1200  14  da. 

18.  $100  53  da, 

622.  To  find  the  interest  at  any  rate  per  annum. 

623.  Examples.     1.   What  is  the  interest  on  $840  for  54  days 


Find  the 

interest 

at  69 

1o  on: 

1. 

Principal 
$900 

Time 

53  da. 

7. 

Principal 

$  775.10 

Time 

17  da. 

2. 

$  287.10 

47 

da. 

8. 

$211 

43 

da. 

3, 

$  1890 

69 

da, 

9. 

$500 

67 

-da. 

4- 

$14.50 

81 

da. 

10. 

$450 

58 

da. 

5. 

$2JL 

91 

da- 

11. 

$  700.80 

126 

da. 

6. 

$59109 

31 

da. 

12. 

$600 

47 

da. 

SOLUTION.     The  interest  at  6  %  is  found  to  be  $  7.56.     Since 
7.560      8  %  is  £  more  than  the  assumed  rate  6  %,  the  interest  at  8  %  is  I 
2.52        more  than  the  interest  at  6%.     Adding  |  of  $7.56  to  itself,  the 
$  10.08        result  is  found  to  be  $  10.08,  or  the  interest  at  8%. 

2.   What  is  the  interest  on  $ 2100  for  180  days  at  5%  ? 


$21.00 
63.00 
10.50 

$52.50 


SOLUTION.  The  interest  at  6  %  is  found  to  be  $  63.  Since 
5  %  is  \  less  than  the  assumed  rate  6  %,  the  interest  at  5  %  is  \ 
less  than  the  interest  at  6  %.  Taking  \  of  $  63  from  itself,  the 
result  is  found  to  be  $52.50,  or  the  interest  at  5%. 


624.   Hence  the  following  rule : 

Add  or  subtract  from  the  interest  at 
itself  as  the  given  rate  is  greater  or  less  than 


such  a  part  of 


DRILL  EXERCISE 

1.  Given  the  interest  at  6%,  how  may  the  interest  at  7%  be 
found? 

SOLUTION.     7  %  is  \  more  than  6  %.     Hence,  the  interest  at  6  %  increased  by 
\  of  itself  is  equal  to  the  interest  at  7  %. 

2.  Formulate  a  short  method  for  changing  6%  interest  to  8% 
interest;  to  5%  interest;  to  4£%  interest;  to  9%  interest;  to  10% 
interest;  to  1\%  interest. 


§§  624-625] 


INTEREST 


239 


3.  If  the  interest  at  6%  is  $60,  what  is  the  interest  at  7%?  at 
5%?  at8%?  at4J%?  at  7*%? 

4.  If  the  interest  at  6%  is  $  240,  what  is  the  interest  at  9%  ?  at 
10%  ?  at  3%  ?  at  41%  ?  at 


625.  General  Principles.  1.  6%  interest  increased  by  ^  of  itself 
equals  7%  interest;  by  ^  of  itself,  7-i-%  interest;  by  \  of  itself,  8% 
interest  ;  by  \  of  itself,  9  %  interest. 

2.  6%  interest  diminished  by  £  of  itself  equals  5%  interest;  by 
J  of  itself,  41%  interest;  by  \  of  itself,  4%  interest. 


6%  interest  may  be  changed  to  10%  interest  by  dividing  by  6  and  moving 
the  decimal  point  1  place  to  the  right  ;  to  12%  interest  by  multiplying  by  2  ;  to 
3%  interest  by  dividing  by  2  ;  to  any  other  rate  of  interest  by  dividing  by  6  and 
multiplying  by  the  required  rate. 


WRITTEN  EXERCISE 


Find  the  interest  on  : 


1.    $  1750  for  15  da.  at  6%.^  "       17f  $  3741.85  for  6  da.  at  7%/ 

18.  $  5178  for  9  da.  at  9%. 

19.  $  732  for  11  da.  at  6%. 

20.  $  1174.51  for  42  da.  at  8%. 

21.  $340  for  70  da.  at  10%.  ' 

22.  $  1478  for  80  da.  at  6%. 
28.  $  2150  for  96  da.  at  4J%. 

24.  $  1200  for  53  da.  at  6%. 

25.  $  1500  for  87  da.  at  7%. 

26.  $420  for  41  da.  at  5%. 

27.  $360  for  81  da.  at  6%. 

28.  $  2347.50  for  18  da.  at  7%. 

29.  >$  1112.49  for  25  da.  at  8%. 

80.  $  1300  for  13  da.  at  6%. 

81.  $  17,000  for  3  da.  at  5J%. 
32.  $  195.50  for  33  da.  at  10%. 


2.  •  $  1125  for  24  da.  at 

8.  $  742.50  for  30  da.  at  6%. 

4.  .  $  900  for  93  da.  at 

5.  $660  for  63  da.  at  8%.    r 

6.  $  136.42  for  33  da.  at  9%.j 
/      7.  $  1000  for  21  da.  at  10%. 

8.  $  2000  for  12  da.  at  5%. 

9.  $  351.23  for  40  da.  at  4J  %. 

10.  $  1368  for  50  da.  at  3%. 

11.  $93.40  for  150  da.  at  0%. 

12.  $  550  for  75  da.  at  7%. 
18.  $  842.50  for  45  da.  at  6%. 

14.  $  800  for  27  da.  at  5%. 

15.  $  1725  for  57  da.  at  9%. 

16.  $  125  for  55  da.  at  6%. 


240  PERCENTAGE   AND   ITS  APPLICATIONS        [§§  625-626 

33.  $  1050  for  43  da.  at  1%.  87.  9  60  for  50  da.  at  5%. 

34.  9 1560  for  44  da.  at  1%%.  ^  *         38.  $  930  for  83  da.  at  6% . 
85.  9  180  for  47  da.  at  6%.  89.  $  750  for  84  da.  at  6%. 
36.  $120  for  49  da.  at  9%.  40.  $550  for  72  da.  at  7%. 

41.  Find  the  total  amount  of  interest  on : 

•  550  for  18  da.  at  6%.  9 250  for  50  da.  at  6%. 

9  810  for  40  da.  at  7%.  9  593.25  for  80  da.  at  7%. 

$1000  for  41  da.  at  1\%.  $1966  for  75  da.  at  5%. 

9  342.50  for  42  da.  at  5%.  $  450  for  83  da.  at  8%. 

$  1362.50  for  45  da.  at  6%.  $  990  for  63  da.  at  6%. 

42.  Find  the  total  amount  of  interest  on : 

$  720  for  9  da.  at  10%.  $  1124  for  15  da.  at  3%. 

9  7500  for  3  da.  at  1%.  $  550  for  45  da.  at  1\% 

$ 216  for  93  da.  at  8%.  $  160  for  27  da.  at  6%. 

$504  for  54  da.  at  6%.  $240  for  31  da.  at  8%. 

$  600  for  4  da.  at  44$.  $  540  for  41  da.  at  9%. 

43.  Find  the  total  amount  of  interest  on : 

$  1452  for  8  da.  at  3%.  $  1400  for  26  da.  at  6%. 

9 1728  for  10  da.  at  6%:  $  1700  for  29  da.  at  8%. 

$  2150.42  for  17  da.  at  7%.  9  1900  for  37  da.  at  7%. 

9  519  for  24  da.  at  8%.  $  2100  for  43  da.  at  6%. 

$  1600  for  23  da.  at  1\%.  $  3100  for  53  da.  at  3%. 

SHORT  METHODS 

626.  Interest  is  a  product  of  which  the  rate  and  time  are  factors. 
Since  the  rate,  being  a  constant  factor,  may  be  ignored,  it  will  be 
observed  that  it  will  make  no  difference  if,  for  convenience,  the  prin- 
cipal in  dollars  and  the  time  in  days  be  interchanged. 

Thus,  the  interest  on  $600  for  93  days  is  the  same  as  the  interest  on  $93  for 
600  days.  Since  the  interest  for  600  days  is  ^  of  the  principal,  ^  of  $93,  or 
$9.30,  is  the  required  interest  on  $600  for  93  days.  The  interest  on  $150  for  88 
days  is  the  equivalent  of  the  interest  on  $88  for  150  days.  Since  150  is  \  of  600, 
the  required  result  may  be  found  by  taking  ^  of  88  and  dividing  the  result  by  4, 
obtaining  $2.20  as  the  required  interest. 


626-628] 


INTEREST 


241 


By  inspection,  find 

1.  $600  for  93  da. 

2.  $300  for  42  da. 

3.  $200  for  66  da, 

4.  $150  for  44  da. 

5.  $120  for  55  da. 

6.  $60  for  89  da. 

7.  $30  for  56  da. 

8.  $20  for  84  da. 

9.  $15  for  124  da, 

10.  $10  for  66  da. 

11.  $6000  for  139  da. 

12.  $  3000  for  142  da.. 


ORAL  EXERCISE 
the  interest  at  6%  on: 
18.  $2000  for  186  da. 

14.  $  1500  for  64  da. 

15.  $1000  for  126  da. 

16.  $750  for  88  da. 

17.  $1200  for  155  da. 

18.  $2400  for  11  da. 

19.  $1800  for  31  da. 

20.  $3600  for  51  da. 

21.  $4200  for  11  da. 

22.  $5400  for  7  da. 

23.  $240  for  21  da. 

24.  $360  for  17  da. 


25.  $420  for  13  da. 

26.  $4200  for  103  da. 

27.  $  3600  for  108  da. 

28.  $  1200  for  39  da. 

29.  $3000  for  145  da. 
SO.  $  1000  for  246  da. 
81.  $  6000  for  159  da. 
32.  $  1800  for  39  da. 
88.  $2400  for  51  da. 
84.  $  7200  for  19  da. 
35.  $  4800  for  17  da. 
86.  $480  for  11  da. 


627.  When  the  rate  is  not  six  per  cent,  many  times  it  is  desirable 
to  increase  or  diminish  the  principal  or  time,  instead  of  the  interest, 
by  the  proper  fraction. 

628.  Examples.     1.  Find  the  interest  on  $  1500  for  84  da.  at  8%. 

SOLUTION.  Since  8%  is  $  more  than  6%,  if 
the  principal  is  increased  by  $  of  itself,  and  the 
interest  computed  for  the  given  time  at  6%,  the 

result  will  be  equal  to  the  interest  at  8%.  Increasing  $1500  by  \  of  itself, 
the  result  is  $2000.  Interchanging  the  dollars  and  days,  the  problem  in  its 
simplest  form  is  equivalent  to  $84  for  2000  days  at  6%.  Since  a  principal 
will  double  itself  in  6000  days,  it  will  yield  an  interest  equal  to  £  of  itself  in  2000 
days.  £  of  84  equals  28,  making  the  required  interest  $28. 

2.   Find  the  interest  on  $799.59  for  45  da.  at  8%. 

SOLUTION.    Since  8%  interest  is  £  more  than  6% 

$  7.9959  =  $  8.        interest,  if  we  increase  the  time  by  £  of  itself  and  com- 
pute the  interest  on  the  principal  for  this  time  at  6  %. 

the  result  will  be  the  interest  at  8  %.  45,  increased  by  $  of  itself,  equals  60.  .01 
of  any  number  is  the  interest  for  60  days.  Hence,  $8  is  the  required  interest. 


242 


PERCENTAGE   AND   ITS  APPLICATIONS         [§§  628-629 


3.   Find  the  interest  on  $844.20  for  80  da.  at 

SOLUTION.     4^%  interest  is  \  less  than  6%  in. 

$  8.4420  =  $  8.44.        terest.    Hence,  if  we  decrease  the  time  by  £  of  itsell 
and  compute  the  interest  on  the  principal  for  the 

remainder  at  6%,  the  result  will  be  the  interest  at  4*  %.  80  days  decreased  by  \ 
of  itself  equals  60  days.  .01  of  the  principal  is  the  interest  for  60  days  at  6%. 
Hence,  the  required  interest  is  $8.44. 


WRITTEN  EXERCISE 


Find  the  interest  on : 

Principal  Time 

1.  $1200.00  79  da. 

2.  $783.60  45  da. 

3.  $425.80  45  da. 

4.  $1600.00  35  da. 
6.  $3200.00  78  da. 
6.  $2700.00  48  da. 


Rate 


Principal 

7.  $799.59 

8.  $111.10 

9.  $2400.00 

10.  $2400.00 

11.  $3800.00 

12.  $1200.00 


Time 

48  da. 
48  da. 
59  da. 
38  da. 
73  da. 
66  da. 


Rate 


Six  Per  Cent  Method 
DRILL  EXERCISE 

1.  What  is  the  interest  on  $  1  for  1  yr.  at  6%  ?    4  yr.  ?    5  yr.  ? 
8yr.? 

2.  What  is  the  interest  on   $10  for  2  yr.  at  6%  ?    on  $30? 
on  $25 ?    on  $80  for  5  yr.  at  6%  ? 

3.  What  is   the   interest  on  $1  for  1  mo.  at  6%  ?   for  2  mo.? 
for  4  mo.  ?  for  6  mo.  ?  for  8  mo.  ?   for  3  mo.  ?  for  7  mo.  ? 

4.  What  is  the  interest  on  $1  for  6  da.  at  6%  ?   for  1  da.  ? 

5.  How  many  mills  will  $1  yield  in  12  da.  at  6%  ?     in  24  da.  ? 
in  9  da.  ?   in  15  da.  ? 

6.  What  is  the  interest  on  $1  for  1  yr.  at  6%  ?   1  mo.  ?   1  da.  ? 

7.  At  6%,  what  is  the  interest  on  $1  for  1  yr.  2  mo.  6  da.? 
on  $20?   on  $2000?  on  $3000?  on  $1500?  on  $7500? 

8.  Give  a  simple  way  to  find  the  interest  on  any  principal  for 
any  given  number  of  years,  months,  and  days. 

629.    General  Principles.     $1  in  1  year  at  6%  will  yield  $.06 
interest ;  in  1  month,  $  .005  interest ;  in  1  day,  $  .000 \  interest. 


§§  630-632] 


INTEREST 


243 


630.  To  find  the  interest  on  any  principal  for  any  time  and  rate  by 
the  six  per  cent  method. 

631.  Example.     What  is  the  interest  on  $650  for  2  yr.  4  mo. 

12  da.  at  6%  ? 

OP  y     2      12  6^)0  SOLUTION.     The  interest  on  $  1  for  1 

.005  x    4     .02  .142 


.0001  x  12     .002      1  300 
.142 


yr.  at  6%  is  $ .06,  and  for  1  mo.,  or  ^  yr., 
it  is  ^  of  $.06,  or  $.005,  and  for  1  da.    ~- 
of  a  month   ^  - 


242 


PERCENTAGE   AND   ITS  APPLICATIONS         [§§  628-629 


S.   Find  the  interest  on  $844.20  for  80  da.  at 


$8.4420  =  $8.44. 


Time           Rate 

Principal 

Time 

79  da.     1\% 

7. 

$799.59 

48  da. 

45  da.     8% 

8. 

$111.10 

48  da. 

45  da.    4$, 

9. 

$2400.00 

59  da. 

35  da.     1\% 

10. 

$2400.00 

38  da. 

78  da.     1\% 

11. 

$3800.00 

73  da. 

48  da.    8% 

12. 

$1200.00 

66  da. 

SOLUTION.  4|%  interest  is  J  less  than  6%  in. 
terest.  Hence,  if  we  decrease  the  time  by  £  of  itsell 
and  compute  the  interest  on  the  principal  for  the 
remainder  at  6%,  the  result  will  be  the  interest  at  4*  %.  80  days  decreased  by  | 
of  itself  equals  60  days.  .01  of  the  principal  is  the  interest  for  60  days  at  6  %. 
Hence,  the  required  interest  is  $8.44. 

WRITTEN  EXERCISE 

Find  the  interest  on : 

Principal         Time  Rate  Principal         Time  Rate 

1.  $1200.00 

2.  $783.60 
8.  $425.80 

4.  $1600.00    35  da.     1\%  10.  $2400.00    38  da,     5% 

5.  $3200.00 


Six  Per  Cent  Method 
DRILL  EXERCISE 

1.  What  is  the  interest  on  $  1  for  1  yr.  at  6%  ?    4  yr.  ?    5  yr.  ? 
8yr.? 

2.  What  is  the  interest  on   $10  for  2  yr.  at  6%  ?    on  $30? 
on  $25  ?    on  $80  for  5  yr.  at  6%  ? 

8.    What  is  the   interest  on  $1  for  1  mo.  at  6%  ?   for  2  mo.? 
for  4  mo.  ?  for  6  mo.  ?  for  8  mo.  ?   for  3  mo.  ?  for  7  mo.  ? 

4.  What  is  the  interest  on  $1  for  6  da.  at  6%  ?   for  1  da.  ? 

5.  How  many  mills  will  $1  yield  in  12  da.  at  6%  ?     in  24  da.  ? 
in  9  da.  ?   in  15  da.  ? 

6.  What  is  the  interest  on  $  1  for  1  yr.  at  6%  ?   1  mo.  ?   1  da.  ? 

7.  At  6%,  what  is  the  interest- on  $1  for  1  yr.  2  mo.  6  da.? 
on  $20?   on  $2000?  on  $3000?  on  $1500?  on  $7500? 

8.  Give  a  simple  way  to  find  the  interest  on  any  principal  for 
any  given  number  of  years,  months,  and  days. 

629.    General  Principles.     $1  in  1  year  at  6%  will  yield  $.06 
interest ;  in  1  month,  $  .005  interest ;  in  1  day,  $  .OOOJ  interest. 


§§  630-632]  INTEREST  243 

630.  To  find  the  interest  on  any  principal  for  any  time  and  rate  by 
the  six  per  cent  method. 

631.  Example.     What  is  the  interest  on  $650  for  2  yr.  4  mo. 

12  da.  at  6%  ? 

06  X    2      12  650  SOLUTION.     The  interest  on  $1  for  1 

yr.  at  6%  is  $.06,  and  for  1  mo.,  or  ^  yr., 
it  is  TL  of  $.06,  or  $.005,  and  for  1  da.,  or 

.0001  x  12     .002       1  300       ^  of  a  month,  it  is  ^  of  $  .005,  or  $  .OOOJ. 
.142     9100          If  the  interest  on  $1  for  1  yr.  is  $.06,  for 
92.300        2  yr.  it  is  twice  $.06,  or  $.12.     If  the  in- 
terest for  1  mo.  is  $.005,  for  4  mo.  it  is  4 

times  $.005,  or  $.02.  If  the  interest  for  1  da.  is  $.000£,  the  interest  for  12  da. 
is  12  times  $.000±,  or  $.002.  If  the  interest  on  $1  for  2  yr.  is  $.12,  for  4  mo. 
$.02,  and  for  12  da  $.002,  the  interest  on  $1  for  2  yr.  4  mo.  12  da.  is  $.142.  If 
the  interest  on  $1  is  $.142,  the  interest  on  $650  is  650  times  $.142,  or  $92.30. 

632.  Hence  the  following  rule  may  be  derived : 

Multiply  the  interest  on  $  1  for  the  given  time  at  6%  by 
the  number  of  dollars  in  the  principal,  and  the  result  is  the 
interest  at  6%. 

Change  6%  to  any  other  rate  of  interest  by  625. 


WRITTEN  EXERCISE 

Find  the  interest  by  the  6%  method. 

Kate 


Principal 

1.  $750.50 

Time 

4  yr.  11  mo. 

Kate 

6% 

Principal 

4.  $1116 

Time 

3  yr.  11  mo. 

&  $3560.00 

9  yr.  10  mo. 

8%  1 

5.  $17,500 

2  yr.  1  mo. 

3.  $610.15 

7  yr.  11  da. 

1% 

6.  $2400 

7  yr.  1  mo. 

7.  On  the  16th  of  September,  1904,  I  borrowed  $3500  at  8%, 
interest.     How  much  will  settle  the  loan  Jan.  1,  1910  ? 

8.  My  note  for  $875.25,  given  2  yr.  9  mo.  27  da.  ago,  bearing 
4%  interest,  is  due  to-day.     What  is  the  amount  of  interest  due  ? 

9.  July  16,  1903,  I  borrowed  $2750  at  5%  interest,  and  on  the 
same  day  loaned  it  at  7-J%  interest.     If  full  settlement  is  made 
Jan.  4,  1905,  how  much  will  be  gained  ? 


244  PERCENTAGE   AND   ITS   APPLICATIONS         [§§  632-635 

10.  Find  the  amount  of  interest  at  6%  by  the  six  per  cent 
method  onr^fc 

$  680,  for  2  y«r.  6  mo.  10  da.  $500,  for  3  yr.  1  mo.  27  da. 

$  1895,  for  1  yr.  7  mo.  7  da  9  895,  for  5  yr.  11  mo.  11  da. 

$468,  for  5  yr.  5  mo.  1  da.  $1650,  for  1  yr.  10  mo.  23  da 

$  1000,  for  11  yr.  1  mo.  20  da  $  1463,  for  9  yr.  1  mo.  9  da. 

$  645,  for  4  yr.  4  mo.  5  da.  $  365,  for  4  y r.  1  mo.  25  da. 

EXACT  INTEREST 

633.  Exact  interest  is  interest  computed  for  the  exact  time  in 
days  on  the  basis  of  365  days  to  a  common  year  and  366  days  to  the 
leap  year     It  is  used  by  the  United  States  government  and  by  a 
few  merchants  and  bankers. 

Aside  from  the  uses  in  government  calculations,  exact  interest  is  rarely  com- 
puted. While  it  is  enforcible,  being  strictly  legal,  the  greater  convenience  of  the 
360-day  rules  so  commend  them  to  public  favor  as  to  lead  to  their  common  use. 

634.  To  change  common  interest  to  exact  interest 

635.  On  a  basis  of  12  periods  of  30  days  each,  a  year's  interest 
is  taken  for  too  short  a  period,  since  a  year,  exclusive  of  a  leap  year, 
contains  365  days.     The  time  is,  therefore,  five  days  or  -g-f-g-,  equal 
to  ^  too  short,  and  the  interest  taken  on  that  basis  is  proportion- 
ately too  great. 

To  correct  this  error  and  obtain  the  exact  interest, 

Subtract  -part  from  any  interest  computed  on  the  360-day  basis. 

WRITTEN  EXERCISE 

Find  the  exact  interest  of: 

1    $954  for  63  days  at  7%.  6.  $681.80  for  90  days  at  10%. 

8.   $630  for  50  days  at  6%.  7.  $500  for  48  days  at  $6%. 

8.   $800  for  33  days  at  5%.  8.  $  1200  for  31  days  at  5%. 

4.   $137.50  for  93  days  at  8%.          9.  $1500  for  55  days  at  1\% 
6.  $210.54  for  100  days  at  9%.      10.   $4500  for  75  days  aft  &%.A b 

11.   $920  from  Apr.  15  to  July  25,  at  6%. 

IS.  $  1756.90  from  May  5  to  Aug.  2,  at  6%. 

IS.   $  2500.75  from  June  25  to  Dec.  8,  at  6%. 

14.  $3200  from  Oct.  15  to  Nov.  25,  at  6%. 

15.  $  2500  from  Apr.  16  to  June  7,  at  6%. 


' 


§§  030-639]  INTEREST  245 

PROBLEMS  IN  INTEREST 

636.  The  four  distinct  elements  considered  in  ilrorest  are  the 
principal,  rate  per  cent,  time,  and  interest  or  amount.    Simce  the  fourth 
element  is  practically  the  product  of  the  first  three,  if  any  three  of 
the  elements  are  given,  the  other  may  be  found  in  accordance  with 

the  general  principles  of  percentage. 

» 

637.  To  find  the  rate  per  cent,  the  principal,  interest,  and  time 
being  given. 

638.  Example.    At  what  rate  per  cent  must  $2100  be  loaned  for 
2  yr.  5  mo.  6  da,  to  gain  $459.90  ? 

SOLUTIOW 

Let  1  %  equal  the  rate. 

$  51.10  =  interest  on  $  2100  for  the  given  time  at  1  %. 

$459.90-*- $51.10  =  9. 

The  interest  at  1  %  is  contained  in  the  given  interest  9  times. 

Therefore  the  required  rate  is  9  times  1  %,  or  9%. 

639.  From  the  above  solution  the  following  rule  may  be  derived : 
Divide  the  given  interest  by  the  interest  an  the  given 

principal  for  the  given  time  at  1%. 

ORAL  EXERCISE 
Find  the  rate  of  interest : 

Principal        Interest          Time  Principal        Interest          Time 

1.  $600        $72        2yr.  4.   $200        $24        4  yr. 

2.  $500        $60        Syr.  6.  $400        $16        6  mo. 
&  $300        $60       5yr.  6.  $100        $24       Syr. 

WRITTEN  EXERCISE 

1.  At  what  rate  will  $  1260  yield  $  13.44  interest  in  96  days  ? 

2.  The  interest  for  $2400  for  1  yr.  8  mo.  6  da.  is  $262.60.    Find 
the  rate  of  interest. 

3.  If  I  pay  $518.75  interest  on  $1250  for  5  yr.  6  mo.  12  da., 
what  is  the  rate  per  cent  ?  *|  V^. 

4.  A  lady  deposited  in  a  savings  bank  $3750,  on  which  she 
received  $93.75  interest  semiannually.     WKat  per  cent  of  interest 
did  she  receive  on  her  monev  ?   ^T  ?, 


246  PERCENTAGE   AND   ITS   APPLICATIONS         [§§640-644 

640.  To  find  the  time,  the  principal,  interest,  and  rate  of  interest 
being  given. 

641.  Example.    In  what  time  at  S%  will  $2000  gain  $400 

interest  ? 

SOLUTION 

Let  1  year  represent  the  time. 

8  %  of  $  2000  =  $  160,  the  interest  on  the  given  principal  for  1  year. 
400  -r-  160  =  2.5. 

Since  the  interest  for  1  year  is  contained  in  the  given  interest  2.6  times,  the 
required  interest  must  be  2.5  times  the  assumed  time. 

1  year  x  2.6  =  2.5  year,  or  2  years  6  months,  the  required  time. 

642.  From  the  above  solution  the  following  rule  may  be  derived : 

Divide  the  given  interest  by  the  interest  on  the  principal 
for  1  year  at  the  given  rate  per  cent. 

ORAL  EXERCISE 

Find  the  time  in  each  of  the  following  problems : 

Principal         Interest         Rate  Principal  Interest  Bate 

L  $700        $84        6%  4.  $750        $   7.50        6% 

2.  $250        $90        4%  d.  $900        $67.50        5% 

8.  $400        $44%  6.  $600        $  6.00        6% 

WRITTEN  EXERCISE 

1.  How  long  will  it  take  $360  to  gain  $53.64  at  5%? 

2.  How  long  should  I  keep  $466.25  at  8%  to  have  it  amount  to 
$610.48?   Vlo'1*/ 

8.  A  debt  of  $1650  was  paid  with  5|%  interest  on  Aug.  30, 1888, 
by  delivering  a  check  for  $2316.85.  At  what  date  was  the  debt 
contracted  ? 

643.  To  find  the  principal,  the  Interest,  rate  of  interest,  and  time 
being  given. 

644.  Example.     What  principal  will  yield  $400  interest  in  2  yr 
6  mo.  at  S%? 


§§  644-647]  INTEREST  247 

SOLUTION 

Let  $  1  represent  the  principal. 
$  .20  =  the  interest  on  $  1  for  2  yr.  6  mo.  at  8%. 
400  -*-  .20  =  2000. 

The  interest  on  the  required  principal  is  2000  times  the  interest  on  the 
assumed  principal. 

Therefore  the  required  principal  is  2000  times  $1,  or  $2000. 

ORAL  EXERCISE 
Find  the  principal  in  each  of  the  following  problems : 

Interest          Kate  Time  Interest          Rate  Time 

1.   $40        6%      6yr.  8  mo.  4.  $24        9%      8  mo. 

&   $42        6%      Syr.  6  ma     .         6.  $32        6%      6  mo.  12  da. 

S.  $50      7i%      240  da.  6.  $50      1\%      24  da. 

WRITTEN   EXERCISE 

1.  What  principal  at  1%  will  gain  $  1080  in  3  yr.  6  mo.  ?  ^ 

2.  What  principal  at  4%  will  yield  $455  in  3  yr.  6  mo.  18  da.  ? 

S.  A  dealer  who  clears  121%  annually  on  his  investment  is 
forced  by  ill  health  to  give  up  his  business.  He  lends  his  money  at 
7%,  by  which  his  income  is  reduced  $1512.50.  How  much  had  he 
invested  in  his  business  ?  ^'J  £~&~d 

645.  To  find  the  principal,  the  amount,  rate  per  cent,  and  time 
being  given. 

646.  Example.    What  principal  will  amount  to  $508  in  4  yr. 

6  mo.  at  6%? 

SOLUTION 

Let  $  1  represent  the  principal. 
$  1.27  =  the  amount  of  $  1  for  4  yr.  6  mo. 
$  508  =  the  amount  of  a  certain  principal  for  4  yr.  6  ma 
$508 -4- $1.27  =  400. 

Since  the  given  amount  is  400  times  the  assumed  amount,  the  required  prin- 
cipal must  be  400  times  the  assumed  principal. 
400  times  $  1  =  $  400,  the  required  principal. 

647.  From  the  above  solution  the  following  rule  may  be  derived : 

Divide  the  given  amount  by  the  amount  of  $  1  for  the 
given  time  and  rate. 


248  PERCENTAGE  AND  ITS  APPLICATIONS  [§§647-650 

ORAL  EXERCISE 

Find  the  principal  in  each  of  the  following  problems : 

Amount  Time  Kate  Amount  Time  Kate 

1.  $1120  2yr.  6%  5.  $1025  3  mo.  10% 

8.  $2080  6  mo.  8%  6.  $1212  60  da.  6% 

S.  $4090  90  da.  9%  7.  $218  2  yr.  4^% 

4.  $3120  6  mo.  8%  8.  $367.50  3  yr.  1\% 

WRITTEN  EXERCISE 

1.  Find  the  principal  that  will  amount  to  $3360  in  3  yr.  at  4%.  - 

2.  What  sum  of  money  put  to-day  at  6%  interest  will  amount  in 
7  mo.  12  da.  to  $4148? 

8.  Owed  a  debt  of  $  5310  due  in  1  yr.  6  mo.  18  da.  I  deposited 
in  a  bank  that  allowed  me  4%  interest  a  sum  sufficient  to  cancel  my 
debt  when  due.  Find  the  sum  deposited. 

4.  I  borrowed  a  certain  sum  for  2  yr.  6  mo.  with  the  understand- 
inff  that  I  was  to  pay  interest  at  the  rate  of  8%.  If  at  maturity  I 
gave  my  check  for  $2400,  what  was  the  sum  loaned  me  ? 

PERIODIC  INTEREST 

648.  Periodic  interest  is  simple  interest  on  the  principal  and  on 
any  interest  remaining  unpaid. 

649.  When  interest  is  payable  annually,  it  is  called  annual 
.interest;  when  payable  semiannually,  semiannual  interest;  when  pay- 
able quarterly,  quarterly  interest;  etc. 

650.  In  some  states  annual  and  other  periodic  interest  is  sanc- 
tioned by  law,  but  in  many  states  it  cannot  be  legally  enforced.     To 
secure  periodic  interest  in  any  state,  it  must  be  specified  by  contract. 

Periodic  interest  is  sometimes  secured  by  a  note  or  a  series  of  notes  ;  in  such 
cases  the  principal  only  is  secured  by  one  of  the  series  (if  not  by  mortgage  or 
otherwise),  while  each  of  the  other  notes  is  drawn  for  one  interest  payment,  and 
matures  on  the  date  at  which  such  payment  is  due.  By  such  arrangement, 
periodic  interest  can  be  enforced  in  states  where  it  would  otherwise  be  regarded 
as  illegal. 


§§  651-655]  INTEREST  249 

651.  To  find  periodic  interest. 

652.  Example.     Find  the  interest  on  $400  for  2  yr.  at  6%,  pay- 
able semiannually. 

SOLUTION 

$48  =  the  simple  interest  for  the  whole  time. 

$12  =  the  semiannual  interest. 

1  yr.  6  mo.  =  the  period  for  which  1st  interest  remained  unpaid. 

1  yr.  =  the  period  for  which  2d  interest  remained  unpaid. 

6  mo.  =  the  period  for  which  3d  interest  remained  unpaid. 

3  yr.  =  the  period  for  which  one  semiannual  interest  draws  interest. 

$2.16  =  the  simple  interest  on  $12  for  3  yr. 

$48  +  $2.16  =  $50.16,  the  semiannual  interest  due. 

653.  From  the  above  analysis  the  following  rule  may  be  derived : 

To  the  simple  interest  on  the  principal  for  the  full  time 
add  the  interest  on  one  period's  interest  for  the  aggregate 
time  for  which  the  payments  of  interest  were  deferred. 

WRITTEN  EXERCISE 

1.  Find  the  quarterly  interest  on  $1600  for  2  yr.  at  6%. 

2.  What  is  the  difference  between  the  simple  and  the  annual 
interest  of  $  2000  for  3.yr.  at  6%  ? 

8.  Find  the  amount  of  interest  due  at  the  end  of  4  yr.  9  mo.  on 
a  note  for  $1155  at  6%,  interest  payable  annually,  but  remaining 
unpaid. 

4.  On  a  note  of  $  1750,  dated  Aug.  1,  1898,  given  with  interest 
payable  annually  at  10%,  the  first  three  payments  were  made  when 
due.  How  much  remained  unpaid,  debt  and  interest,  Jan.  1, 1905  ? 

COMPOUND  INTEREST 

654.  Compound  interest  is  the  interest  on  the  principal  and  on 
the  principal  increased  by  the  interest  at  the  expiration  of  regular 
intervals. 

655.  Interest  may  be   added  to  the  principal  annually,  semi- 
annually,, or  quarterly,  according  to  agreement. 

Compound  interest  is  not  recoverable  by  law,  but  a  creditor  may  receive  it 
if  tendered  without  incurring  the  penalty  of  usury  ;  and  a  new  obligation  may 


250  PERCENTAGE   AND  ITS  APPLICATIONS         [§§655-658 

also  be  taken  at  the  maturity  of  a  compound  interest  claim  for  the  amount  so 
shown  to  be  due,  and  such  new  obligation  will  be  valid  and  binding. 

Most  savings  banks  allow  compound  interest  on  balances  remaining  on 
deposit  for  a  full  interest  term. 

656.  To  find  compound  interest 

657.  Exampla     Find  the  compound  interest  on  $600  for  3  yr. 

6  ino.  at  6%. 

SOLUTION 

$600  x  $  .06          =  $  86.00,  the  interest  for  the  first  year. 

$  600  +  $  36  =  $  636. 00,  the  amount  for  the  first  year. 

$636  x  $.06          =  $  38.16,  the  interest  for  the  second  year. 

$  636  +  $  38.16       =  $  674.16,  the  amount  for  the  second  year. 

$674.16  x  $  .06      =  $  40.45,  the  interest  for  the  third  year. 

$674.16  +  $40.45  =  $714.61,  the  amount  for  the  third  year. 

$  714.61  x  $  .03      =  $   21.44,  the  interest  for  6  mo. 

$714.61  +  $21.44  =  $736.05,  the  amount  for  the  full  time. 

$  736.05  —  $  600     =  $  136.05,  the  compound  interest  for  the  full  time. 

658.  Hence  the  following  rule  inay  be  derived  : 

Find  the  amount  of  the  principal  and  interest  for  the 
first  period  and  make  that  the  principal  for  the  second 
period,  and  so  proceed  to  the  time  of  settlement. 

Subtract  the  principal  from  the  last  amountf  and  the 
remainder  will  be  the  compound  interest. 

If  the  time  contains  fractional  parts  of  a  period,  find  the  amount  due  for  the 
full  periods,  and  to  this  add  its  interest  for  the  fractional  period. 


WRITTEN  EXERCISE 

1.  What  is  the  compound  interest  on  $1200  for  4  years  at  7% 
if  the  interest  is  compounded  annually  ?  Vt  ^ '  ^ 

2.  What  is  the  compound  interest  on  $600  for  3  years  at  5%  if 
the  interest  is  compounded  quarterly  ? 

8.   Find  the  compound  interest  on  $ 400  for  4  years  at  4%, 
interest  payable  semiannually. 

4.  Find  the  amount  at  compound  interest  on  $  500  for  3  yr.  4  mo. 
at  5%,  interest  payable  annually. 

5.  Find  the  compound  interest  at  6%  on  $2000  for  1  yr.  5  mo., 
interest  payable  quarterly. 


§658] 


INTEREST 


251 


COMPOUND  INTEREST  TABLE 


Showing  the  amount  of  $1  at  compound  interest  at  various 
rates  per  cent  for  any  number  of  years,  from  1  year  to  50  years, 
inclusive. 


Yrs. 

1  per  ct. 

Hperct 

2  per  ct. 

2^  per  ct. 

3  per  ct. 

3£  per  ct. 

4  per  ct. 

1 

1.0100000 

1.0150000 

1.02000000 

1.02500000 

1.03000000 

1.03500000 

1.04000000 

2 

1.0201  000  11.0302  250 

1.0404  0000 

1.0506  2500 

i.ot;o90000 

1.07122500 

1.08160000 

3 

1.0303010 

L0456784 

1.0612  0800 

1.0768  9062 

1.0927  2700 

1.1087  1787 

1.12486400 

4 

1.0406  040 

1.0613  636 

1.08243216 

.1038  1289 

1.1255  0881 

1.14752300 

1.16985856 

5 

1.0510  101 

1.0772  840 

1.1040  8080 

.1314  0821 

1.1592  7407 

1.1876  8631 

1.21665290 

6 

1.0615202 

1.0934433 

1.1261  6242 

.1596  9342 

1.19405230 

1.22925533 

1.2653  1902 

7 

1.0721  354 

1.1098450 

1.1486  8567 

.1886  8575 

1.22987387 

1.2722  7926 

1.31593178 

8 

1.0828567   1.1264926 

1.1716  5938 

.2184  0290 

1.26677008 

1.31680904 

1.3685  6905 

9 

1.0936  853 

1.1433900 

1.19509257 

.2488  6297 

1.30477318 

1.3628  9735 

1.42331181 

10 

1.1046  221 

1.1605408 

1.2189  9442 

.2800  8454 

1.3439  1638 

1.4105  9876 

1.48024428 

11 

1.1156683 

1.1779489 

1.24337431 

.3120  8666 

1.38423387 

1.45996972 

1.53945406 

12 

1.1268250 

1.1956  182 

1.2682  4179 

.3448  8882 

1.42576089 

1.51106866 

1.60103222 

13 

1.1380933 

1.213.-)  524 

1.2936  0663 

.3785  1104 

1.46853371 

1  .5639  5606 

1.66507351 

14 

1,1494742 

1.2317  557 

1.3194  7876 

.4129  7382 

1.5125  8972 

1.6186  9452 

1.73167(545 

15 

1.1609  690 

1.2502  321 

1.34586834 

.4482  9817 

1.5579  6742 

1.67534883 

1.80094351 

16 

1.1725  786 

1.2689  855 

1.3727  8570 

.4845  0562 

1.6047  0644 

1.73398601 

1.87298125 

17 

1.1843044 

1.2880203 

1.4002  4142 

.5216  1826 

1.65284763 

1.79467555 

1.94790050 

18 

.1961  475 

1.3073406 

1.4282  4625 

.5596  5872 

1.70243306 

1.857485)20 

2.02581(552 

19 

.2081  090 

1.3269507 

1.4568  1117 

.5086  5019 

1.75350605 

1.92250132 

2.10(i84918 

20 

.2201  900 

1.3468  550 

1.48594740 

1.63&6  1644 

1,8061  1123 

1.9897  8886 

2.1911  2314 

21 

1.2323919 

1.3670578 

1.51566634 

.6795  8185 

1.8C029457 

2.05943147 

2.2787  6807 

22 

.2447  159 

1.3875637 

1.5459  7967 

.7215  7140 

1.91610341 

2.1315  1158 

2.3699  1879 

23 

.2571  630 

1.4083772 

1.57689926 

.7646  1068 

1.97358651 

2.2061  1448 

2.4647  1555 

24 

1.2697346 

1.4295  028 

1.6084  3725 

.8087  2595 

2.0327  9411 

2.2833  2849 

2.5633  0417 

25 

1.2824  320 

1.4509454 

1.64060599 

.8539  4410 

2.0937  7793 

2.3632  4498 

2.6658  3633 

26 

1.2952563 

1.4727095 

1.6734  1811 

.9002  9270 

2.1565  9127 

2.4459  5856 

2.7724  6979 

27 

1.3082089 

1.4948  002 

1.7068  8648 

.9478  0002 

2.2212  8901 

2.5315  6711 

2.8833  6858 

28 

1.3212  910 

1.5172222 

1.7410  2421 

.9964  9502 

2.2879  2768 

2.6201  7196 

2.9987  0332 

29 

1.3345039 

1  .5399  805 

1.77584469 

2.0464  0739 

2.3565  6551 

2.7118  7798 

3.11865145 

30 

1.3478490 

1.5630802 

1.81136158 

2.0975  6758 

2.4272  6247 

2.8067  9370 

3.24339751 

31 

1.3613274 

1.5865  264 

1.84758882 

2.1500  0677 

2.5000  8035 

2.9050  3148 

3.3731  3341 

32 

1.3749407 

1.6103243 

1.88454059 

2.2037  5694 

2.5750  8276 

3.0067  0759 

3.5080  5875 

33 

1.3886901 

1.6344792 

1.92223140 

2.2588  5086 

2.6523  3524 

3.11194235 

3.64838110 

34 

1.4025  770 

1.6589964 

1.9606  7603 

2.3153  2213 

2.73190530 

3.2208  6033 

3.7943  1634 

35 

1.4166028 

1.6838  813 

1.99988955 

2.3732  0519 

2.81386245 

3.3335  9045 

3.9460  8899 

36 

1.4307688 

1.7091  395 

2.0398  8734 

2.4325  3532 

2.8982  7833 

3.4502  6611 

4.10393255 

37 

1.4450765 

1.7347766 

2.0806  8509 

2.49334870 

2.9852  2668 

3.5710  2543 

4.2680  8986 

38 

1.4595272 

1  .7607  983 

2.1222  9879 

2.5556  8242 

3.0747  8348 

3.6960  1132 

4.4388  1345 

39 

1.4741225 

1.7872  103 

2.1647  4477 

2.6195  7448 

3.1670  2698 

3.8253  7171 

4.6163  6599 

40 

1.4888637 

1.8140184 

2.2080  3966 

2.6850.6384 

3.2620  3779 

3.9592  5972 

4.8010  2063 

41 

1.5037524 

1.8412  287 

2.25220046 

2.7521  9043 

3.3598  9893 

4.0978  3351 

4.9930  6145 

42 

1.5187899 

1.8688471 

2.2972  4447 

2.8209  9520 

3.4606  9589 

4.2412  5799 

5.1927  8391 

43 

1.5339778 

1.8968  798 

2.3431  8936 

2.8915  2008 

3.5645  1677 

4.38970202 

5.40049527 

44 

1.5493  176 

1.9253330 

2.39005314 

2.9638  0808 

3.6714  5227 

4.5433  41(50 

5.6165  1508 

45 

1.5648107 

1.9542  130 

2.4378  5421 

3.03790328 

3.7815  9584 

4.7023  5855 

5.8411  7568 

46 

1.5804589 

1.9835262 

2.4866  1129 

3.1138  5086 

3.89504372 

4.86694110 

6.0748  2271 

47 

1.5962634 

2.0132  791 

2.5:5634351 

3.19169713 

4.01  18  9503 

5.0372  8404 

6.3178  1562 

48 

1.6122261 

2.0434  7S3 

2.5870  7039 

3.2714  8956 

4.13225188 

5.21358898 

6.5705  2824 

49 

1.6283483 

2.0741  305 

2.63881179 

3.35327680  4.25621944 

2.  30(50  6459 

6.8333  4937 

50 

1.6446318 

2.1052424 

2.6915  8803 

3.43710872  4.38390602 

5.58492686 

7.10668335 

252 


PERCENTAGE   AND   ITS   APPLICATIONS 


[§658 


COMPOUND  INTEREST  TABLE  ' 

Showing  the  amount  of  $1  at  compound  interest,  at  various 
rates  per  cent,  for  any  number  of  years,  from  1  year  to  50  years, 
inclusive. 


Yrs. 

4£  per  ct.     5  per  ct. 

6  per  ct. 

7  per  ct. 

8  per  ct. 

9  per  ct. 

10  per  ct. 

1 

1.04500000 

1.0500  000 

1.0600000 

1.0700000 

1.0800000 

1.0900000 

1.1000000 

2 

1.09202500 

1.1025000 

1.1236000 

1.1449000 

1.1664000 

1.1881  000 

1.2100000 

3 

1.14116612 

1.1576  250 

1.1910  160 

1.2250430 

1.2597  120 

1.2950290 

1.3310000 

4 

1.19251860 

1.2155063 

1.2624770 

1.3107960 

1.3604  890 

1.4115816 

1.4641  000 

5 

1.2461  8194 

1.2762  816 

1.3382256 

1.4025517 

1.4(593281 

1.5386  240 

1.6105  100 

6 

1.3022  6012 

1.3400956 

1.4185  191 

1.5007  304 

1.5668743 

1.6771  001 

1.7715610 

7 

1.36086183- 

1.4071  004 

1.5036303 

1.6057  815 

1.7138  243 

1.8280391 

1.9487  171 

8 

1.4221  0061 

1.4774554 

1.5938481 

1.7181  862 

1.8509302 

1.9925626 

2.1435  888 

9    1.4860  9514 

1.5513282 

1.6894  790 

1.8384  592 

1.9990046 

2.1718  933 

2.3579  477 

10 

1.55296942 

1.6288946 

1.7908477 

1.9671  514 

2.1589250 

2.3673637 

2.5937  425 

11    1.62285305 

1.7103394 

1.8982986 

2.1048  520 

2.3316  390 

2.5804  264 

2.8531  167 

12    1.69588143 

1.7958563 

2.0121  965 

2.2521  916 

2.5181  701 

2.8126  648 

3.1384284 

13    1.77219610 

1.8856491 

2.1329283 

2.4098  450 

2.7196  237 

3.0658046 

3.4522  712 

14  1  1.8519  4492 

1.9799316 

2.2609  040 

2.5785  342 

2.9371  936 

3.3417  270 

3.7974  983 

15    1.93528244 

2.0789  282 

2.3965582 

2.7590315 

3.1721691 

3.6424  825 

4.1772482 

16  12.02237015 

2.1828  746 

2.5403  517 

2.9521  638 

3.4259426 

3.9703059 

4.5949  730 

17    2.11337681 

2.2920  183 

2.6927  728 

3.1588  152 

3.7000  181 

4.3276334 

5.0544  703 

18  j  2.2084  7877 

2.4066  192 

2.8543  392 

3.3799323 

3.9960  195 

4.7171  204 

5.5599  173 

19 

2.30786031 

2.5269  502 

3.0255  995 

3^165  275 

4.3157011 

5.1416613 

6.1159390 

20 

2.4117  1402 

2.6532  977 

3.2071  355 

3^8696  845 

4.6(509571 

5.6044  108 

6.7275000 

21 

2.5202  4116 

2.7859626 

3.3995636" 

'4.1405  624 

5.0338  337 

6.1088077 

7.4002499 

22 

2.6336  5201 

2.9252607 

3.6035  374 

4.4304017 

5.4365404 

6.6586004 

8.1402749 

23 

2.7521  6635 

3.0715  238 

3.8197  497 

4.7405299 

5.8714  637 

7.2578  745 

8.9543024 

24 

2.87601383 

3.2250  999 

4.0489  346 

5.0723  670 

6.3411  807 

7.9110  832 

9.8497  327 

25 

3.00543446 

3.3863549 

4.2918  707 

5.4274  326 

6.8484  752 

8.6230807 

10.8347  059 

26 

3.1406  7901 

3.5556  727 

4.5493  830 

5.8073  529 

7.3963532 

9.3991  579 

11.9181765 

27 

3.28200956 

3.7334  563 

4.8223459 

6.2138  676 

7.9880615 

10.2450  821 

13.1099942 

28 

3.42969999,   3.9201291 

5.1116  867 

6.6488  384 

8.6271064 

11.1671  395 

14.4209  936 

29 

3.58403649    4.1161356 

5.4183  879 

7.1142571 

9.3172  749 

12.1721  821 

15.8630  930 

30 

3.7453  1813 

4.3219424 

5.7434912 

7.6122550 

10.0626  569 

13.2676785 

17.4494023 

31 

3.9138  5745 

4.5380  395 

6.0881  006 

8.1451  129 

10.8676694 

14.4617  695 

19.1943425 

32 

4.08998104 

4.7649  415 

6.4533867 

8.7152  708 

11.7370  830 

15.7633  288 

21.1137768 

33 

4.2740  3018 

5.0031  885 

6.8405899 

9:3253  398 

12.6760  496 

17.1820284 

23.2251  544 

34 

4.46636154 

5.2533480 

7.2510  253 

9.9781  135 

13.6901  336 

18.7284  109 

25.5476  699 

35 

4.66734781 

5.5160  154 

7.6860  868 

10.6765  815 

14.7853443 

20.4139  679 

28.1024369 

36 

4.8773  7846 

5.7918  161 

8.1472520 

11.4239422 

15.9681  718 

22.2512250 

30.9126805 

37 

5.0968  6049 

6.0814069 

8.6360  871 

12.2236  181 

17.2456  256 

24.2538  353 

34.0039486 

38 

5.3262  1921 

6.3854773 

9.1542  524 

13.0792  714 

18.6252  756 

26.4366  805 

37.4043434 

39 

5.5658  9908 

6.7047  512 

9.7035  075 

13.9948  204 

20.1152977 

28.8159817 

41.1447778 

40 

5.81636454 

7.0399887 

10.2857  179 

14.9744  578 

21.7245  215 

31.4094  200 

45.2592  556 

41 

6.0781  0094 

7.3919  882 

10.9028610 

16.0226699 

23.4624832 

34.2362679 

49.7851  811 

42 

6.3516  X548 

7.7615  876 

11.5570327 

17.1442568 

25.3394  819 

37.3175  320 

54.7636  992 

43 

6.6374  3818 

8.1496669 

12.2504546 

18.3443548 

27.3666.404 

40.6761  098 

60.2400  692 

44 

6.9361  2290 

8.5571  503 

12.9854  819 

19.6284596 

29.5559  717 

44.3369  597 

66.2640  761 

45 

7.24824843 

8.9850078 

13.7646  108 

21.0024518 

31.9204494 

48.3272  861 

72.8904837 

46 

7.5744  1961 

9.4342  582 

14.5904  875 

22.4726  234 

34.4740  853 

52.6767  419 

80.1795  321 

47 

7.9152  6849 

9.9059711 

15.4659  167 

24.0457  070 

37.2320  122 

57.4176486 

88.1974853 

48 

8.2714  5557 

10.4012  697 

16.3938  717 

25.7289065 

40.2105  731 

(52.5852  370 

97.0172  338 

49 

8.6436  7107 

10.9213  331 

17.3775040 

27.5299300 

43.4274  190 

68.2179083 

106.7189572 

50 

9.0326  3627 

11.4(573998 

18.4201  543 

29.4570  251 

46.9016  125 

74.:;r>7520i 

117.3908529 

§§  059-660]  INTEREST  253 

Application  of  Compound  Interest  Table 

659.  To  find  the  amount  of  any  given  principal  for  any  given  num- 
ber of  years  : 

Multiply  the  given  principal  by  the  amount  of  $1  at  the  given  rate, 
as  shown  by  the  table. 

For  periods  beyond  the  scope  of  the  table,  multiply  together  the  amounts 
shown  for  periods,  the  sum  of  which  will  equal  the  time  required. 

For  example,  to  find  the  compound  amount  of  $  1  for  100  years  : 

Multiply  the  amount  for  50  years  by  itself;  to  find  the  amount  for  75  years, 
multiply  the  amount  for  50  years  by  the  amount  for  25  years;  etc. 

If  interest  is  to  be  compounded  semiannually,  take  one  half  the  rate  for  twice 
the  time.  If  interest  is  to  be  computed  quarterly,  take  one  quarter  of  the  rate 
for  4  times  the  time  /  etc. 

660.  To  find  the  compound  interest  on  principals  of  $  100  or  less,  use 
four  of  the  decimal  places  shown  in  the  table;  on  principals  of  $1000 
or  less,  use  five  of  the  decimal  places  ;  and  so  on. 


WRITTEN   EXERCISE 

By  the  use  of  the  compound  interest  table  solve  the  following 
problems  : 

1.  Find  the  amount  of  $  1750  compounded  annually  for  10  yr. 
6  mo.  at  5%. 

2.  Find  the  compound  interest  on  $800  for  16  yr.  4  mo.  18  da. 
at  8%,  interest  compounded  annually. 

8.   Find  the  compound  interest  at  6%  on  $500  for  1  yr.  3  mo., 
interest  payable  quarterly. 

4.  Find  the  compound  interest  atx8%  on  $1200  for  2  yr.  5  mo., 
interest  payable  quarterly. 

5.  What  principal  will,  in  8  yr.  at  5%,  amount  to  $4107.26,  if 
interest  is  compounded  semiannually  ? 

ORAL  REVIEW 

1.   What  principal  in  2  yr.  at  6%  will  yield  $24  interest?    $72 
interest  ?     $  84  interest  ? 

2-   What  principal  in  8  yr.  at  5%  will  amount  to  $280? 


254  PERCENTAGE  AND  ITS  APPLICATIONS  [§660 

8.  What  sum  of  money  invested  at  6%  will  in  2  yr.  3  mo.  yield 
$  135  interest  ?  $  405  interest  ? 

4-   At  what  rate  would  $  300  in  3  y r.  yield  $  45  interest  ? 

&  A  lady  deposited  in  a  savings  bank  $750,  on  which  she 
received  $15  interest  semiannually.  What  per  cent  of  interest  did 
she  receive  on  her  money  ? 

6.  To  satisfy  a  debt  of  $400  which  had  been  standing  2  yr.,  I 
gave  my  check  for  $440.     What  was  the  rate  of  interest  charged? 

7.  How  long  will  it  take  $100  to  gain  $12  at  6%  ? 

8.  How  long  must  $  550  be  on  interest  at  7  %  to  amount  to  $  570  ? 

9.  In  what  time  will  money  bearing  8%  interest  double  itself? 
SOLUTION.     In  order  to  double  itself  the  interest  accumulated  must  be  equal 

to  100%  of  the  principal.  Since  the  principal  increases  8%  per  annum,  it  will 
require  as  many  times  one  year  to  increase  100%,  or  to  double  itself,  as  8%i* 
contained  times  in  100%,  or  12£  times,  equal  to  12  yr.  6  mo. 

10.  In  what  time  will  any  sum  bearing  interest  at  4%  double 
itself?  at  5%  ?  at  10%  ?  at  9%  ? 

11.  How  many  days  will  $6000  require  to  yield  $71  interest? 

12.  In  how  many  days  will  the  interest  at  6%  be  one  half  of  the 
principal  ?  double  the  principal  ? 

WRITTEN  REVIEW 

1.  A  note  of  $  1260  is  151  days  past  due.    What  amount  will 
settle  the  note  and  interest,  money  being  worth  6%  per  annum  ? 

2.  The  interest  on  a  certain  sum  in  12 \  years  is  \  of  that  sum. 
What  is  the  rate  of  interest?  Lf-  H  0 

3.  A  certain  principal  placed  at  simple  interest  for  64  days 
amounts   to   $606.40.      If   the    same   principal   would   amount   to 
$624.90  in  249  days,  what  is  the  rate  of  interest?    What  is  the 
principal  ? 

4-  What  monthly  rent  should  be  charged  for  a  house  costing 
$10,240  in  order  that  6%  interest  on  the  investment  may  be 
realized  ?  £\  .  3^0 

5.  At  the  end  of  five  years  the  accrued  interest  on  a  certain 
principal  is  found  to  be  £  of  the  sum  drawing  interest.  What  is  the 

rate  of  interest  ? 

H     JO 


§660]  INTEREST  255 

6.  A  merchant  charges  interest  at  the  rate  of  6  %  per  annum  on 
overdue  accounts.     He  received  a  check  for  $  1205.40  in  settlement 
for  an  overdue  account   of   $1200.      How  long   overdue  was   the 
account?  ^.1   ibk    . 

7.  A  man  borrowed  $12,000  at  5%  and  with  it  immediately 
bought  a  house  which  he  rented  for  $1800  per  year.     What  was 
his  yearly  per  cent  of  net  gain  or  loss  ?    In  how  many  years  will  his 
net  gains  aggregate  the  sum  borrowed  ? 

8.  May  16  I  bought  300  barrels  of  flour,  at  $7  per  barrel; 
July  28  I  sold  50  barrels,  at  $8  per  barrel ;  Oct.  30,  100  barrels,  at 
$6.75  per  barrel ;  and  Feb.  13  following,  the  remainder,  at  $7.80  per 
barrel.     Allowing  interest  at  6%,  what  was  my  gain  ? 

9.  Find  the  simple  interest  on  £40  8s.  5d.  for  2  years  at  6%. 
Give  the  answer  in  United  States  money. 

10.  Find  the  compound  interest,   by  the  table,   on  $2000  for-^x 
4  years  at  6%,  interest  payable  quarterly.          5^S\-  ^\\ 

11.  A  man  invested  $  16,000  in  business,  and  at  the  end  of  3  yr. 
3  mo.  withdrew  $  22,240,  which  sum  included  investment  and  gains. 
What  yearly  per  cent  of  interest  did  his  investment  pay  ?       \  0^    I  f\ 

12.  Find  the  interest  of  that  sum  for  11  yr.  8  da.  at  10i$f, 
which  will,  at  the  given  rate  and  time,  amount  to  $  1715.08. 

18.  Sold  an  invoice  of  crockery  on  a  credit  of  2  months;  the 
bill  was  paid  3  mo.  18  da.  after  the  date  of  purchase,  with  interest 
at  8%,  by  a  check  for  $  1963.45.  How  much  was  the  interest  ? 

14.  A  man  having  $  21,000  invested  it  in  real  estate,  from  which 
he  received  a  semiannual  income  of  $  787.50.     He  sold  this  property 
at  cost  and  invested  the  proceeds  in  a  business  which  yielded  him 
$  472.50  quarterly.     How  much  greater  rate  per  cent  per  annum  did 
he  receive  from  the  second  investment  than  from  the  first  ? 

15.  In  order  to  engage  in  business,  I  borrowed  $3750  at  6%, 
and  kept  it  until  it  amounted  to  $4571.25.     How  long  did  I  keep 
the  money  ?  \-~  ^ 

16.  In  what  time  will  interest  at  8%  equal  f  of  the  principal  ? 

17.  A  building  which  cost  $  10,500  rents  for  $  87.50  per  month. 
What  annual  rate  of  interest  on  his  investment  does  the  owner 
receive  if  he  pays  yearly  taxes  amounting  to  $102.50;  insurance, 
$21.25;  repairs,  $136.80;  and  janitor's  services,  $  56.95  ? 


256  PERCENTAGE   AND   ITS  APPLICATIONS         [§§  660-666 

18.  A  merchant  sold  a  stock  of  glassware  on  one  month's  credit; 
the  bill  was  not  paid  until  3  mo.  21  da.  after  it  became  due,  at  which 
time  the  seller  received  a  draft  for  $4716.21  for  the  bill  and  interest 
thereon  at  the  rate  of  5%.     Find  the  selling  price  of  the  goods. 

19.  Oct.  12,  1904,  I  purchased  2700  bushels  of  wheat  at  $1.05 
per  bushel,  and  afterwards  sold  it  at  a  profit  of  6%.     On  what  date 
was  the  wheat  sold  if  my  gain  was  equivalent  to  10%  interest  on 
my  investment  ? 

20.  I  am  offered  a  house  that  will  rent  for  $  27  per  month,  at 
such  a  price  that,  after  paying  $  67.20  taxes  and  other  yearly  expenses 
amounting  to  $  24.85,  my  net  income  will  be  8J%  on  my  investment. 
What  is  the  price  asked  for  the  house  ? 

PRESENT  WORTH  AND  TRUE  DISCOUNT 

661.  The  present  worth  of  a  debt  payable  at  a  future  time  with- 
out interest  is  such  a  sum  as  being  put  at  simple  interest  at  a  legal 
rate  will  amount  to  the  given  debt  when  it  becomes  due. 

662.  The  true  discount  is  the  difference  between  the  face  of  the 
debt  due  at  a  future  time  and  its  present  worth. 

To  illustrate,  suppose  A  owes  B  $212  to  be  paid  for  one  year  after  date. 
Should  A  care  to  cancel  the  indebtedness  at  once,  the  sum  which  he  ought  to  pay 
should  be  such  that,  if  put  out  at  legal  interest  by  B,  it  would  at  the  end  of  the 
year  amount  to  $  212. 

Suppose  that  B  can  receive  6  %  on  his  money.  At  this  rate  $  1  put  at  simple 
interest  would,  at  the  end  of  one  year,  amount  to  $  1.06.  If  $  1  in  one  year 
amounts  to  $  1.06,  it  will  take  as  many  times  $  1  to  amount  to  $  212  as  $  1.06  is 
contained  times  in  $  212,  or  200  times.  Hence,  $200  must  be  the  sum  which  A 
ought  to  pay  now  to  cancel  a  debt  which  at  the  end  of  one  year  amounts  to 
$212. 

The  $  200  to  be  paid  is  called  the  present  worth,  and  the  difference  between 
$212  and  $200,  or  $  12,  the  true  discount. 

663.  Computations  in  present  worth  and  true  discount  come 
under  the  case  of  interest  problems,  in  which  the  amount,  the  rate 
per  cent,  and  the  time  are  given,  to  fin<J  the  principal  or  interest. 
The  debt  corresponds  to  the  amount;  the  rate  per  cent  agreed  upon 
to  the  rate  ;  the  time  intervening  before  the  maturity  of  the  debt  to 
the  time ;  the  present  worth,  which  is  the  unknown  term,  to  the 
principal ;  and  the  true  discount  to  the  interest. 


§§  664-G66J       PRESENT   WORTH   AND   TRUE  DISCOUNT  257 

664.  To  find  the  present  worth  and  true  discount  of  a  debt. 

665.  Example.     Find  the  present  worth  and  true  discount  of  a 
debt  of  $545  payable  in  1  yr.  6  mo.,  when  money  may  be  loaned 

at  6% 

SOLUTION 

Let  $  1  represent  the  present  worth  of  the  debt. 
$0.09  =  the  interest  on  $  1  for  1  yr.  6  mo.  at  6  %. 
$1.09  =  the  amount  of  $  1  for  1  yr.  6  mo.  at  6  %. 
$  545  •*-$  1.09  =  600. 

Since  the  present  worth  of  $1.09  is  $1,  the  present  worth  of  $645,  which  is 
600  times  $1.09,  must  be  500  times  $1,  or  $500. 
$545  -  $500  =  $45,  the  true  discount. 

666.   Hence  the  following  rule  may  be  derived : 

Divide  the  amount  of  the  debt  at  maturity  by  the  amount 
of  $1  for  the  given  time  and  rate  and  the  quotient  will  be 
the  present  worth. 

Subtract  the  present  worth  from  the  amount  of  the  debt 
and  the  remainder  will  be  the  true  discount. 

ORAL  EXERCISE 

L   Find  the  true  discount  on  $  330  due  2  years  hence  at  5%. 

2.  What  principal  is  that  which  in  2  years  at  5%  will  amount 
to  $210? 

8.  What  is  the  present  worth  of  $  224  to  be  paid  in  2  years  if 
the  money  is  loaned  at  6%  ? 

4-  Which  is  the  better,  and  how  much,  to  buy  a  piano  for  $  636 
on  12  months'  time,  or  to  pay  $  580  cash,  money  being  worth  6%? 

5.  If  money  is  worth  6  %,  what  cash  offer  will  be  equivalent  to 
an  offer  of  $  102.50  for  a  bill  of  goods  on  5  months'  credit  ? 

6.  A  merchant  marks  an  article  with  two  prices,  one  for  cash, 
$48,  and  the  other  for  credit  of  6  months,  $  51.50.     If  money  is 
worth. 6  %,  which  is  the  better  price  for  the  buyer?  for  the  seller  ? 

WRITTEN  EXERCISE 

1.  Which  is  greater,  and  how  much,  the  interest  or  the  true 
discount  on  $516  due  in  1  year  8  months,  if  money  is  worth  10  % 
per  annum  ? 


258  PERCENTAGE   AND  ITS  APPLICATIONS 

2.  Which  is  the  better,  and  how  much,  to  buy  flour  at  $6.75  per 
barrel  on  6  months'  time,  or  to  pay  $  6  cash,  money  being  worth  6%? 

3.  When  money  is  worth  5%  per  annum,  which  is  preferable, 
to  sell  a  house  for  $40,000  cash,  or  for  $42,000,  due  in  1  year  ? 

4.  A.  farmer  offered  to  sell  a  pair  of  horses  for  $  420  cash,  or 
for  $475  due  in  15  months  without  interest.    If  money  is  worth  8% 
per  annum,  how  much  would  the  buyer  gain  or  lose  by  accepting  the 
latter  offer  ? 

5.  If  money  is  worth  6%,  what  cash  offer  will  be  equivalent  to 
an  offer  of  $  1546  on  a  bill  of  goods  on  90  days'  credit  ?   /S%  J/«) 

6.  An  agent  paid  $  840  for  a  traction  engine,  and  after  holding 
it  in  stock  for  1  year,  sold  it  for  $933.80  on  8  months'  credit.     If 
money  was  worth  6%,  what  was  his  actual  gain  ?    *J 'tff 

7.  Find  the  difference  between  the  interest  and  true  discount 
on  $4160  for  1  year  at  4%. 

8.  A  stock  of  moquette  carpeting  bought  at  $  1.95  per  yard  on 
8  months'  credit  was  sold  for  cash  on  the  day  "of  purchase  at  $  1.80 
per  yard.     If  money  was  worth  6%  per  annum,  what  per  cent  of 
gain  or  loss  did  the  seller  realize  ? 

9.  William  is  now  16  years  old.     How  much  money  must  be 
invested  for  him  at  6%  simple  interest  in  order  that  he  may  have 
$  19,500  principal  and  interest  when  he  celebrates  his  21st  birthday? 

10.  What  amount  of  goods  bought  on  6  months'  time,  or  5%  off 
for  cash,  must  be  purchased  in  order  that  they  may  be  sold  for 
$4180  and  net  the  purchaser  10%  profit  on  paying  cash  and  getting 
the  agreed  discount  off  ? 

11.  A  merchant  bought  a  bill  of  goods  for  $2150  on  6  months' 
credit  and  the  seller  offered  to  discount  the  bill  5%  for  cash.     If 
money  is  worth  1\%  per  annum,  how  much  would  the  merchant 
gain  by  accepting  the  seller's  offer  ? 

12.  On  Jan.  1,  1903,  I  paid  $800  for  a  debt  of  $832  payable  on 
a  certain  future  date.     If  money  was  worth  S%  per  annum  and  my 
payment  was  the  present  worth  of  the  claim,  when  was  the  debt 
due? 

18.  What  amount  of  goods  bought  on  4  months'  time,  10%  off  if 
paid  in  1  month,  5%  off  if  paid  in  2  months,  must  be  purchased  in 
order  that  they  may  be  sold  for  $11,480  and  |  of  the  stock  net 


§§666-670]  NEGOTIABLE   PAPER  259 

a  gain  of  W%,  and  the  remainder  a  gain  of  20%  to  the  purchaser, 
if  he  pays  his  bill  within  1  month,  and  gets  the  agreed  discount 
off? 

14.  The  true   discount  on  $4160  for  1  year   is  $160.     What 
is  the  rate  of  interest  ? 

15.  I  sold  my  farm  for  $  10,000,  the  terms  being  one  fifth  cash 
and  the  remainder  in  four  equal  semiannual  payments  with  simple 
interest  at  5  %  on  each  from  date.    Three  months  later  the  purchaser 
settled  in  full  by  paying  with  cash  the  present  worth  of  the  deferred 
payments  on  the  basis  of  10%  for  the  use  of  the  money.     How 
much  cash  did  I  receive  in  all? 

NOTE.     Before  finding  the  present  worth  of  interest-bearing  debts,  add  to 
the  face  of  the  debt  the  simple  interest  for  the  time  to  the  date  of  maturity. 

NEGOTIABLE  PAPER 

667.  Negotiable  paper  is  paper  which  may  be  transferred  from 
one  owner  to  another  by  indorsement  or  delivery,  or  both ;  as,  prom- 
issory notes,  drafts,  and  bills  of  exchange. 

668.  A  promissory  note,  or  a  note,  is  a  written  promise  on  the 
part  of  one  person  to  pay  another  a  certain  sum  of  money  on  demand 
or  at  a  certain  specified  future  time. 

669.  A  negotiable  note  is  one  which  is  made  payable  to  the  order 
of  a  designated  payee,  or  to  the  bearer. 


BOTTOM,  MASS., 


^^ 


aftrr  date^promise  to  pay  to 


the  order  of 


•£%  DOLLARS 


''  yW^zz^z^gx 


VALUE  RECEIVED. 

Nn.   2.6T.     Due 


670.   The  parties  whose  names  appear  on  a  note  when  it  is  made 
are  the  maker  and  the  payee. 


260  PERCENTAGE   AND   ITS   APPLICATIONS         [§§671-676 

671.  The  maker  is  the  person  who  promises  to  pay  and  whose 
name  is  signed  to  the  note.     The  payee  is  the  person  named  in  the 
body  of  the  note  and  the  one  to  whom  the  money  is  to  be  paid. 

When  a  note  is  made  payable  to  a  certain  person,  or  bearer,  or  to  the  bearer, 
any  person  who  is  the  lawful  holder  of  the  note  is  the  payee. 

672.  The  face  of  a  note,  draft,  etc.,  is  the  sum  for  which  it  is  given. 

In  the  foregoing  form,  John  P.  Kennedy  is  the  maker  ;  Charles  M.  Eastman, 
the  payee  ;  and  $500,  the  face  of  the  note. 

When  no  place  of  payment  is  named,  a  note  is  payable  at  the  residence  or 
place  of  business  of  the  maker. 

When  a  note  contains  the  words  "  with  interest"  or  "  with  use"  and  does 
not  specify  the  rate,  it  draws  interest  at  the  legal  rate  for  the  place  where  it  is 
drawn  up.  A  note  containing  an  interest  clause  will  bear  interest  from  its  date 
unless  other  time  be  specified.  When  the  words  "with  interest"  or  "with 
use"  are  omitted,  the  note  does  not  draw  interest,  and  is  called  non-interest- 
bearing.  Non-interest-bearing  notes  become  interest-bearing  after  maturity. 

673.  A  non-negotiable  note  is  one  that  is  drawn  payable  only  to 
the  party  named  in  the  note. 

If  the  words  "to  the  order  of"  were  omitted  in  the  foregoing  form,  the 
note  would  be  non-negotiable,  and  not  transferable  except  by  assignment. 

674.  A  draft  is  a  written  order  by  one  person  on  another  for 
the  payment  of  a  specified  sum  of  money  to  a  third  person,  or  to 
his  order. 

675.  With  reference  to  time  of  payment  thero  are  two  kinds  of 
drafts,  —  sight  drafts  and  time  drafts. 

676.  A  sight  draft  is  one  drawn  payable  "at  sight";  that  is, 
when  it  is  presented  to  the  drawee  for  payment. 


BUFFALO,  N.  Y.v 


& 


.Pay  to  the  order  of 


VALUE  RECEIVED,  and  charge  to  the  account  of 


^DOLLARS, 


§§677-679]  NEGOTIABLE   PAPER  201 

677.   A  time  draft  is  one  payable  on  a  specified  date,  a  certain 
time  after  date,  or  a  certain  time  after  sight. 


No. 


678.  The  drawer  is  the  person  who  writes  or  draws  the  draft. 
The  drawee  is  the  person  on  whom  the  draft  is  drawn.     The  payee  is 
the  person  to  whom  the  draft  is  to  be  made  payable. 

In  the  above  draft  Fred  A.  Fernald  is  the  drawer;  "W.  P.  Emerson,  the 
drawee  ;  and  James  W.  Mace,  the  payee. 

A  draft,  being  simply  an  order  to  pay  money,  is  not  binding  on  the  drawee 
without  his  consent. 

By  accepting  the  above  draft  W.  P.  Emerson  has  expressed  his  willingness 
to  pay  it.  The  draft  is  then  an  acceptance,  and  W.  P.  Emerson  the  acceptor. 

The  drawee's  liability  begins  with  his  acceptance  of  the  draft.  By  accepting 
the  above  draft  W.  P.  Emerson  has  practically  created  a  promissory  note  with 
himself  as  maker. 

The  acceptance  on  drafts  payable  after  sight  should  always  be  dated  to  make 
it  possible  to  determine  the  maturity  of  the  paper. 

A  draft  payable  "  after  date "  begins  to  mature  from  the  date  of  the 
paper. 

A  draft  payable  "  after  sight "  begins  to  mature  from  the  date  of  the  accept- 
ance of  the  paper. 

To  honor  a  draft  is  to  accept  it  or  pay  it  upon  presentation. 

679.  An  indorsement  is  that  which  is  written  on  the  back  of  a 
note  or  bill  which  pertains  to  the  transfer  or  payment  thereof. 

The  person  who  indorses  or  writes  his  name  on  the  back  of  a  note  or  bill  is 
called  the  indorser,  and  the  person  to  whom  the  note  or  bill  is  indorsed,  the 
indorsee. 


PERCENTAGE   AND  ITS  APPLICATIONS        [§§  680-683 


FULL  INDORSEMENT 


680.  The  purchaser  of  negotiable  paper  has  a  valid  claim  against 
the  original  and  subsequent  parties  thereto  if  he  can  show: 

1.  That  he  gave  value  for  it. 

2.  That  he  bought  it  before  maturity. 

3.  That  to  the  best  of  his  knowledge  and  belief  no  fraud  was 
connected  with  it. 

681.  Indorsements  are  made  on  notes  and  drafts  for  three  pur- 
poses:   1.   To  secure  their  payment.    2.  To  effect  their  transfer 
3.   To  make  memorandum  of  a  partial  payment. 

682.  The  principal  forms  of  indorsement  in  common  use  in  business 
transactions  are :   1.   Full.     2.   Blank.     3.    Qualified. 

683.  A   full    indorse- 
ment   is    one    in  which 
the  indorser  directs  the 
payment   of   the  instru- 
ment to  be  made  to  the 
order,   of     a     particular 
party.     A  blank  indorse- 
ment is  one  in  which  the 
indorser    merely    writes 
his  name  on   the  back, 
thus  making  the  instru- 
ment    payable     to     the 
bearer.     A    qualified    in- 
dorsement is  one  in  which 
the  indorser  relieves  him- 
self  from   the    ordinary 
responsibility   of  an   in- 
dorser by  placing  over  his 
name  the  words  "without 
recourse."      A   qualified 
indorsement  may  be   in 
blank  or  in  full. 

For  greater  security, 
checks,  notes,  and  drafts 
should  generally  be  indorsed 
in  full.  A  check,  note,  or 
draft  used  for  remittance  purposes  should  always  be  indorsed  in  full 


BLANK  INDORSEMENT 


QUALIFIED  INDORSEMENT 


O  u 


§§683-691]  BANK  DISCOUNT  263 

When  paper  is  indorsed  in  blank,  the  lawful  holder  of  it  may,  if  he  chooses, 
write  over  the  name  of  the  indorser  the  words  necessary  to  convert  the  blank 
indorsement  into  a  full  indorsement. 

684.  A  protest  is  a  written,  or  partly  written  and  partly  printed, 
declaration  made  by  a  notary  public,  of  the  demand  and  refusal  to 
pay  or  accept  negotiable  paper. 

685.  A  note  is  dishonored  if  the  maker  refuses  payment  at 
maturity ;  a  draft,  if  the  drawee  does  not  accept  or  pay  it  upon 
presentation. 

If  the  holder  of  a  note  or  draft  fails  to  give  notice  of  dishonor  by  the  maker 
or  acceptor,  the  indorsers  of  the  paper  are  not  liable  for  its  non-payment. 

BANK  DISCOUNT 

686.  A  commercial  bank  is  a  corporation  chartered  by  law  for  the 

receiving  and  loaning  of  money,  for  facilitating  its  transmission,  and, 
in  case  of  banks  of  issue,  for  furnishing  a  circulating  medium. 

687.  Bank  discount  is  a  deduction  from  the  sum  due  upon  a 
negotiable  paper  at  its  maturity  for  the  cashing  or  buying  of  such 
paper  before  it  becomes  due. 

688.  In  true  discount  the  present  worth  is  taken  as  the  principal ; 
in  bank  discount  the  future  worth  is  taken  as  the  principal. 

689.  The  time  in  bank  discount  is  the  period  included  from  the 
date  of  the  paper  to  its  maturity. 

690.  The  term  of  discount  is  the  time  from  the  date  of  discount 
to  the  date  of  maturity. 

Banks  generally  charge  interest  for  the  exact  number  of  days. 

In  a  few  cities  both  the  day  of  date  and  day  of  maturity  are  included  in  the 
term  of  discount. 

In  this  text  the  term  of  discount  is  found  by  taking  the  exact  number  of 
days  from  the  date  of  discount  to  the  date  of  maturity. 

691.  The  discount  is  the  interest  at  the  legal  rate  for  the  term 
of  discount ;  it  is  always  paid  in  advance. 

A  fixed  sum  may  be  taken  as  the  discount,  but  this  is  unusual. 


264  PERCENTAGE   AND  ITS  APPLICATIONS         [§§  692-695 

692.  The  maturity  of  negotiable  paper  is  the  date  upon  which 
it  is  due. 

The  maturity  of  a  note  or  draft  payable  after  date  is  determined  by  adding 
the  time  to  the  date  of  the  note  or  draft ;  the  maturity  of  a  draft  payable  after 
sight  by  adding  the  time  to  the  date  of  the  acceptance.  When  the  time  of  a  note 
or  draft  is  expressed  in  months,  simply  the  number  of  months,  regardless  of  the 
number  of  days  included,  is  counted  ;  but  when  the  time  is  expressed  in  days, 
the  exact  number  of  days,  regardless  of  the  number  of  months  included,  is 
counted  ;  e.g.  a  note  or  draft  due  two  months  from  date  matures  on  the  corre- 
sponding day  of  the  second  month  following,  providing  there  is  a  sufficient  num- 
ber of  days  in  that  month  to  make  this  possible.  If  there  are  not  days  enough, 
the  note  or  draft  is  due  on  the  last  day  of  that  month.  Thus,  notes  and  drafts 
dated  Jan.  28,  29,  30,  and  31,  and  payable  one  month  after  date,  would  mature 
Feb.  28 ;  but  if  payable  30  days  after  date,  they  would  mature  Feb.  27,  28, 
Mar.  1,  and  Mar.  2,  respectively. 

In  a  great  many  of  the  states  the  added  day  for  leap  year  is  not  counted, 
and  a  note  or  draft  payable  30  days  after  Feb.  25  in  any  leap  year  would  be 
considered  due  Mar.  27  instead  of  Mar.  26,  as  would  be  the  case  if  the  extra  day 
were  counted.  In  this  text  the  extra  day  for  leap  years  is  counted. 

693.  The  proceeds  of  a  note  or  other  negotiable  paper  is  the 
amount  paid  to  the  one  offering  it  for  discount ;  it  is  equal  to  the 
face  of  the  paper  less  the  discount. 

694.  The  value  of  a  note  at  its  maturity  is  its  face  if  the  note 
does  not  bear  interest;  and  its  face  plus  the  interest  for  the  time 
if  the  note  does  bear  interest. 

695.  Days  of  grace  are  the  three  days  sometimes  allowed  by  law 
for  the  payment  of  a  note  or  time  draft  after  the  expiration  of  the 
time  specified  in  the  note  or  draft. 

Days  of  grace  have  been  abolished  in  so  many  states  that  all  problems  in 
this  text  omit  them. 

The  laws  of  the  different  states  with  regard  to  paper  maturing  on  Sunday  or 
a  legal  holiday  are  not  uniform.  Some  states  provide  that  when  notes  or  drafts 
mature  on  Sunday,  they  shall  be  paid  on  Saturday,  and  that  if  Saturday  be  a 
legal  holiday,  then  they  shall  be  paid  on  Friday  ;  also  that  if  a  legal  holiday 
occurs  on  Monday,  payment  must  be  made  on  the  preceding  Saturday.  In  other 
states  if  notes  or  drafts  fall  due  on  Sunday  or  on  a  legal  holiday,  they  must  be 
paid  on  the  next  succeeding  business  day,  etc.  The  laws  of  the  different  states 
should  be  carefully  studied  and  fully  observed  in  order  that  all  contingent  parties 
may  be  held  responsible, 


§696]  BANK  DISCOUNT  265 

696.  Computations  in  bank  discount  may  be  made  in  accordance 
with  the  general  principles  of  percentage  or  interest,  the  value  of 
the  note  corresponding  to  the  base  in  percentage,  or  the  principal  in 
interest ;  the  per  cent  of  discount  to  the  rate  or  the  per  cent  of  inter- 
est ;  the  bank  discount  to  the  percentage  or  the  interest ;  and  the 
proceeds  to  the  difference. 

ORAL  EXERCISE 

1.  What  is  the  maturity  of  a  note  dated  Oct.  31  and  payable  in 
60  days?  in  90  days? 

SOLUTION.  Since  60  days  is  equal  to  2  months  of  80  days  each,  to  find  the 
maturity  of  a  note  due  60  days  from  Oct.  31,  count  ahead  two  months,  obtain- 
ing Dec.  31  as  the  due  date  ;  but  as  December  has  31  days  instead  of  80,  to  find 
the  actual  due  date  deduct  this  one  day  from  the  date  of  maturity,  obtaining 
Dec.  30  as  the  result. 

Following  this  line  of  reasoning,  when  the  time  is  90  days  count  ahead 
three  months,  and  the  due  date  is  Jan.  31  ;  but  since  two  of  the  months,  January 
and  December,  each  have  an  extra  day,  deduct  two  days  from  the  approximate 
due  date,  obtaining  Jan.  29  as  the  actual  date  of  maturity. 

By  inspection,  find  the  maturity  of  each  of  the  following  notes : 

Date  Time  Date  Time 

8.  Jan.  29, 1903  1  ma  7.   Sept.  30, 1903  60  da. 

3.  Jan.  30, 1904  1  ma  8.  Oct.     8,1904  90  da. 

4.  Jan.  28,  1903  1  ma  9.  Apr.     5, 1903  1  ma 
£,  Jan.  31, 1904  30  da.  10.  May  30,1904  60  da. 
6.  Oct.  31, 1903  1  mo.  11.  Apr.  15, 1904  6  mo. 

By  inspection,  find  the  maturity  of  the  following  time  drafts : 

Time  after  Time  after  Time  after 

Date  Date  Date  Date  Date  Date 

18.  Mar.  5    30  da.       14.   Sept.  1    2  mo.       16.   Jan.  1      3  mo. 
IS.  July  16  90  da.       15.   May  31    1  mo.       17.  Dec.  28    60  da. 

By  inspection,  find  the  maturity  of  each  of  the  following  drafts : 

Date        Time  after  Date         Time  after  Date     Time  after 

accepted         Sight  accepted          Sight  accepted       Sight 

18.  Aug.  9     10  da.       80.  July  16    90  da.       88.   July  3     3  mo. 

19.  Dec.  12    Imo.       81.  Nov.  30    30  da.       23.  Feb.  28   1  mo 


266  PERCENTAGE  AND   ITS   APPLICATIONS         [§§  696-698 


WRITTEN   EXERCISE 


Find    the   maturity   and   term    of   discount   in   the   following 
problems : 


a/x/^ 

r 

r*i 

L& 

i. 

2. 
3. 
4- 

Date  of 
Note 

Jan.  15 
Mar.  8 
Apr.  7 
Mar.  20 

Time 

3  mo. 
30  da, 
90  da. 
2  mo. 

Date 
discounted 

Jan.  25^ 
Mar.  15 
May  3  ^ 
Apr.  1  lf< 

6. 

6. 
7. 
8. 

Date  of 

Note 

Sept.  20 
June  5 
Aug.  8 
Oct.  1 

Time  v 

3  mo. 
60  da. 
30  da. 

JT 

3  mo. 

»A~A 

Date 
discounted 

Sept.  20 
June  18 
Aug.  15 
Oct.  20 

ft 

«7 

1* 
73 

Find    the   maturity   and   term    of   discount   in   the    following 
problems : 


**  '  Date  of  Draft 

9.   Feb.  12 

Time  after  Sight 

10  da. 

When  accepted 

Feb.  12 

When  discounted 

Feb.  13 

10.   Oct.  6 

30  da. 

Oct.  7 

Oct.  12 

11.  Dec.  24 

60  da. 

Dec.  29 

Dec.  29 

18.  Nov.  30 

3  mo. 

Dec.  1 

Dec.  7 

Find    the    maturity   and   term  of   discount  in   the   following 
problems : 

Date  of  Draft             Time  after  Date  When  accepted  When  discounted 

18.  Mar.  3       90  da.  Mar.  4  Mar.  8 

14.  June  29      3  mo.  July  1  July  10 

15.  Jan.  15      30  da.  Jan.  17  Jan.  17 

16.  Feb.  14      1  mo.  Feb.  15  Feb.  16 

697.   To  find  the  bank  discount  or  proceeds  of  a  note,  the  face,  time, 
and  rate  per  cent  being  given. 


I.   Examples.     1.   Find  the  bank  discount  and  proceeds  of  a 
note  for  $  638  due  in  54  days. 

SOLUTION 

Face  =  $638. 

Bank  discount  =  the  simple  interest  on  the  face  for  54  days,  or  $5.74. 

Proceeds  =  $638  -  $5.74,  or  $632.26. 


§§  G98-699]  BANK   DISCOUNT  26? 

2.   A  note  for  $  540,  dated  Jan.  10,  1903,  payable  in  3  months, 
with  interest  at  6%,  is  discounted  Feb.  27  at  6%.     Find  the  bank 

discount  and  proceeds. 

SOLUTION 
Face  =  $540. 
Interest  =  $8.10. 

Value  of  the  note  at  maturity  =  $540  +  $8.10,  or  $548.10. 
Bank  discount  =  the  interest  on  $548.10  for  42  days,  or  $3.84. 
Proceeds  =  $548.10  -  HL84,  or  $544.26. 


699.   Hence  the  following  rule  may  be  derived : 

Compute  the  interest  on  the  value  of  the  note  for  the  time 
and  rate,  and  the  result  is  the  bank  discount. 

Subtract  the  bank  discount  from  the  value  of  the  note,  and 
the  result  is  the  proceeds. 

WRITTEN  PROBLEMS 
Find  the  bank  discount  and  proceeds  of  the  following  notes : 


Face 

1.  $2400 
2.   $1200 
3.   $3250 

Date  of  Note 

Apr.  2 
Apr.  8 
June  15 

Time 

60  da. 

ft 
2  mo.  ~ 

60  da. 

Date  of 
discount 

Apr.  18 
Apr.  20 
June  30 

Kate  of        /  '? 
discount 

6% 
6% 

4.   $500 
5.   $2500 

Mar.  20 

May  8 

2  mo. 
3  mo. 

Apr.  5 
June  16 

5% 
6% 

6.   $3600 
7.   $1500 

July  8 

Aug.  5 

90  da. 
60  da. 

Aug.  7 
Aug.  5 

6%  ':!  ''" 

8.   $2400 

Sept.  5 

2  mo. 

Sept.  6 

%% 

P.    $4500 

Oct.  2 

30  da. 

Oct.  8 

±\% 

m   $2400 

Nov.  1 

30  da. 

Nov.  5 

8% 

•11.  Apr.  18,  J.  M.  Cox  &  Co.  borrowed  of  National  Bank  of  Re- 
demption $1200  on  their  note  at  60  days,  indorsed  by  W.  C. 
Williams.  Write  the  note  and  show  the  indorsement.  Find  the 
proceeds,  the  rate  of  discount  being  6%. 


268  PERCENTAGE   AND   ITS  APPLICATIONS 

Find  the  date  of  maturity,  the  term  of  discount,  the  discount, 
and  the  proceeds  of  the  following  notes  and  drafts : 

12.    $1200.00.  BOSTON,  MASS.,  Apr.  15,  1903. 

Ninety  days  after  date  I  promise  to  pay  to  the  order  of  Smith 
Bros.  &  Co.,  Twelve  Hundred  Dollars. 

Value  received.  THOMAS  BROWN,  JR. 

Discounted  May  1  at 


+&**-  //" 

IS.    $4500.00.  BOSTON,  MASS.,  Mar.  3,  1903. 

Ninety  days  after  date  we  promise  to  pay  to  the  order  of  E.  M. 
Williams  &  Co.,  Forty-five  Hundred  Dollars  at  National  Bank  of 
Redemption.  Value  received. 

CWA  I  fc  o  *U.  "  *  CHARLES  E.  COLE. 

FRANK  E.  SPRING. 
Discounted  Apr.  2  at  6%. 

14.  $900.00.  DENVER,  COLO.,  Nov.  25,  1903. 
Three  months  after  date  I  promise  to  pay  to  the  order  of  Smith 

Bros.  &  Co.,  Nine  Hundred  Dollars  at  the  Erie  County  National 
Bank.     Value  received. 

W.  L.  SUNDERLAND. 

Discounted  Jan.  1, 1904,  at  8%. 

15.  $660.90.  NEW  ORLEANS,  La.,  May  5,  1903. 
Ninety  days  after  date  I  promise  to  pay  to  the  order  of  H.  H. 

Douglas,  Six  Hundred  Sixty  and  •££$  Dollars,  with  interest  at  6%. 
Value  received.  CLAYTON  S.  MEYERS.   .n 

Discounted  June  1  at  5%.  ^  C    1  4  '-      I  *L>*> 

16.  $2400.00.  BT.  PAUL,  MINN.,  Aug.  31  ,-1903. 
Six  months  after  date  I  promise  to  pay  to  the  order  of  John  W. 

Bell,  Twenty-four  Hundred  Dollars,  with  interest  at  8%,  after  one 
month.     Value  received. 

OLIVER  JONES. 
Discounted  Sept.  5,  1903,  at  8%. 

17.  $  800.00.  CLEVELAND,  O.,  Jan.  31,  1903. 
One  month  after  date  we  promise  to  pay  to  the  order  of  Hale  & 

Bly,  Eight  Hundred  Dollars,  with  interest  at  5%. 

Value  received.  HART  &  COLE. 

Discounted  Feb.  10,  1903,  at  6%. 


\ 


§699] 


BANK   DISCOUNT 


269 


18.  $1200.00.  SPRINGFIELD,  MASS.,  Mar.  5,  1903. 
Four  months  after  date  we  jointly  and  severally  promise  to  pay 

to  the  order  of  Shaw  Bros.  &  Co.,  Twelve  Hundred  Dollars,  with 
interest  from  date.     Value  received.         RAYMOND  D.  DANN, 

Discounted  May  1  at  ^\%.  CHARLES  L.  KINSLEY.    /  2- 

19.  $2576.25.  CHICAGO,  ILL.,  May  20,  1903. 
At  ninety  days'  sight  pay  to  the  order  of  Ourselves,  Twenty-five 

Hundred  Seventy-six  and  -ffa  Dollars,  value  received  and  charge 
to  the  account  of 

To  SPEAR  BROS.  &  Co.,  F.  E.  KoGERS  &  Co. 

San  Francisco,  Cal. 

Accepted  May  25, 1903.     Discounted  May  30,  1903,  at  6%. 


"4-f/^i* 

It&s 


270  PERCENTAGE   AND   ITS  APPLICATIONS  [§699 


22.    $  795j%.  ROCHESTER,  N.  Y.,  May  15,  1904.  ' 

Sixty   days   after  date   pay  to  the  order  of   Ourselves,  Seven 

Hundred  Ninety-five  and  -ffo  Dollars,  value  received,  and  charge 

to  the  account  of 

BOWEN,  MERRILL  &  Co. 

To  GRAY  &  SALISBURY,  Buffalo,  N.  Y. 

Accepted  May  20,  1904.     Discounted  May  21,  1904,  at  6%.     Col- 
lection charges 


28.  A  note  for  $1200,  dated  Boston,  Mass.,  Mar.  4,  1903,  and 
payable  3  months  after  date,  was  discounted  Apr.  5  at  5%.  Find 
the  bank  discount  and  proceeds. 

$4.  What  is  the  proceeds  of  a  note  for  $  3500,  dated  Feb.  2,  1903, 
and  due  in  4  months,  without  interest,  if  the  note  is  discounted 
Apr.  20,  at  4%  ? 

25.  Find  the  date  of  maturity,  'the  bank  discount,  and  proceeds 
of  a  note  for  $  1800,  dated  Feb.  18,  1903,  payable  in  90  days,  and 
discounted  Apr.  25,  1903  at  6%. 

26.  Paul  Harmon's  bank  account  is  overdrawn  $3596.11.    He 
now  discounts  at  6%  :  a  90-day  note  for  $450,  a  60-day  note  for 
$1754.81,  a  30-day  note  for  $851.95,  a  20-day  note  for  $345.25,  a 
10-day  note  for  $100;   proceeds  of  all  to  his  credit  at  the  bank. 
What  is  the  condition  of  his  bank  account  after  he  receives  these 

credits?     \  -        (XQ 

*  vv 

27.  C.  H.  Good  &  Co.'s  bank  account  is  overdrawn  $7,564.19. 
They  discount  at  6%  :  a  90-day  note  for  $  3975.21,  a  60-day  note  for 
$1546.19,  and  a  20-day  note  for  $2546.85;  proceeds  of  all  to  their 
credit  at  the  bank.     What  is  the  condition  of  their  bank  account 
after  they  receive  credit  as  above  ?  v  ,  ,       £ 

28.  Assuming  that  the  model  note,  page  259,  was  discounted 
March  4,  1903  at  6%,  find  the  bank  discount  and  net  proceeds. 

29.  Assuming  that  the  model  draft,  page  261,  was  discounted 
March  9,  1903  at  5%,  find  the  bank  discount  and  net  proceeds; 
collection  charges,  £%. 


§§  700-702]  BANK   DISCOUNT  271 

700.  To  find  the  face  of  a  note,  the  proceeds,  time,  and  rate  of  dis- 
count being  given. 

701.  Example.     What  must  be  the  face  of  a  note,  payable  in  30 
days,  in  order  that  when  discounted  at  6%  the  proceeds  will  be 
$  895.50? 

SOLUTION 

Let  $  1  represent  the  face  of  the  note. 

$  .005  =  the  bank  discount  on  $  1  for  30  da. 

$1  -  $.005  =  $.995,  the  proceeds  of  $1  due  in  30  days. 

$895.50-7- $.995  =  900. 

The  given  proceeds  is  900  times  the  proceeds  of  $  1. 

Therefore  the  required  face  is  900  times  $1,  or  $900. 

702.  Hence  the  following  rule : 

Divide  the  proceeds  of  the  note  by  the  proceeds  of  $  1  at 
the  given  rate  for  the  given  time. 

* 

WRITTEN  EXERCISE 

1.  I  wrsh  to  borrow  $  650.08  of  a  bank.     For  what  sum  must  I 
issue  a  90-day  note  to  obtain  the  amount,  discount  being  at  6%? 

2.  A  man  wishes  to  borrow  $  594  cash.     For  how  much  must  he 
draw  a  note,  so  that  when  discounted  at  6%  on  60  days'  time,  with- 
out interest,  the  proceeds  will  be  the  sum  wanted  ? 

8.  A  30-day  6%  interest-bearing  note  was  discounted  10  days 
after  it  was  drawn  up.  If  the  rate  of  discount  was  6%  and  the  bank 
discount  $  13.40,  what  was  the  face  of  the  note  ? 

4.  A  merchant  bought  goods  to  the  amount  of  $  2376.    For  how% 
much  must  he  draw  his  60-day  note,  without  interest,  that  when  dis- 
counted at  6%  he  may  pay  for  his  purchase  with  the  proceeds  ? 

5.  A  note  dated  Mar.  15,  1903,  payable  in  3  months  with  inter- 
est at  7i%,  was  discounted  Apr.  16,  1903,  at  10%.     If  the  proceeds 
were  $  2404,25,  what  must  the  face"  have  been  ? 

6.  You  have  $650.80  to  your  credit  at  a  bank;  you  give  your 
check  for   $1872.40,  after  which  you  discount  a  30-day  note  for 
$850.80,  proceeds  to  your  credit  at  the  bank.     You  then  discount  a 
90-day  note,  made  by  H.  C.  Davis,  proceeds  to  your  credit,  when 
you  find  yourself  indebted  to  the  bank  $  24.74.     If  discount  be  at 
6%,  what  must  have  been  the  face  of  the  note  made  by  Davis  ? 


272  PERCENTAGE   AND   ITS   APPLICATIONS          [§§  703-706 

PARTIAL  PAYMENTS 

703.  A  partial  payment  is  a  payment  of  a  part  of  the  amount  due 
on  a  note,  mortgage,  or  other  interest-bearing  obligation. 

Partial  payments  should  be  acknowledged  by  indorsement  on  the 
back  of  a  note,  as  follows : 


1 

! 

6 
u 

^g 

T- 

3 

! 

s 

1 

H 

02 

Ci      O 
**H       ^ 
*.      ^> 

^   - 

I 

3 

M 

02 

i 

s 

0 

| 

w 

w 

s^  ~ 
,g  ,g 

\ 

M 

1 

^ 

1 

| 

Sometimes  special  receipts  are  given  for  partial  payments  on  notes  and  other 
similar  obligations. 

A  debtor,  or  his  authorized  agent,  may  make  a  payment,  either  in  part  or  in 
full,  of  any  obligation,  and  such  payment  may  be  received  by  the  creditor,  or 
his  authorized  agent. 

704.  Various  rules  are  in  use  for  finding  the  balance  due  on 
claims  on  which  partial  payments  have  been  made,  but  only  the 
United  States  Rule  and  the  Merchants'  Rule  have  more  than  local 
application. 

UNITED   STATES  RULE 

705.  The  United  States  Rule  is  very  generally  used  when  partial 
payments  are  made  on  interest-bearing  obligations  that  run  for  more 
than  one  year.     It  is  the  rule  that  has  been  adopted  by  the  Supreme 
Court  of  the  United  States,  and  by  most  of  the  separate  states. 

706.  General  Principles.     The  United  States  Eule  recognizes  the 
following  general  principles : 

1.  Accrued  interest  must  be  paid  before  the  principal  may  be 
diminished. 

2.  Interest  must  not  be  charged  upon  interest. 


§§  707-708] 


PARTIAL   PAYMENTS 


273 


707.  To  find  the  balance  due  by  the  United  States  Rule  for  partial 
payments. 

708.  Example.     A  note,  the  face  of  which  was  $2500,  bearing 
interest  at  6%,  was  given  Nov.  1,  1899,  and  settled  Aug.  5,  1904. 
Find  the  balance  due,  the  following  payments  having  been  made : 
Dec.  5,  1900,  $  600 ;  Jan.  5, 1902,  $  500 ;  May  1, 1903,  $  100 ;  July  5, 

1904,  $800. 

SOLUTION 

Face  of  note $2500.00 

Interest  from  Nov.  1, 1899,  to  Dec.  5, 1900  (1  yr.  1  mo.  4  da.)  164.17 

Amount  due  Dec.  5,  1900,  time  of  first  payment       .        .  2664.17 

Payment  of  Dec.  5,  1900 600.00 

New  principal,  or  amount  to  draw  interest  from  Dec.  5, 1900,  2064. 17 

Interest  from  Dec.  5,  1900,  to  Jan.  5,  1902  (1  yr.  1  mo.)     .  '  134.17 

Amount  due  Jan.  5,  1902,  time  of  second  payment  .        .  2198.34 

Payment  Jan.  5,  1902 500.00 

New  principal,  or  amount  to  draw  interest  from  Jan.  5, 1902  1698.34 
Interest  from  Jan.  5,  1902,  to  May  1,  1903  (1  yr.  3  mo. 

26  da.) $134.73 

The  interest  exceeds  the  payment,  and  a  new  principal  is 

not  formed. 
Interest  from  May  1,  1903,  to  July  5,  1904  (1  yr.  2  mo. 

4  da.) 120.02 

Total  interest  due  July  5,  1904 254.75 

Amount  due  July  5,  1904,  the  time  of  the  fourth  payment  1953.09 

Sum  of  payments  May  1,  1903,  and  July  5,  1904               .  9QO.OO 

New  principal,  or  amount  to  draw  interest  from  July  5, 1904  1053.09 

Interest  from  July  5,  1904,  to  Aug.  5,  1904  (1  mo.)  .         .  5.27 

Amount  due  Aug.  5,  1904,  the  final  date  of  settlement     .  1058.36 

CONDENSED  FORM  FOR  WRITTEN  WORK 


DATES 

INTEREST  PERIODS 

PRINCIPALS 

INTERESTS 

AMOUNTS 

PAYMENTS 

Yr.       Mo.  Da 

1899     11     1 

1900      12     5 

1  yr.  1  mo.  4  da. 

$2500.00 

$164.17 

$2664-17 

$600 

1902       7     5 

1        1          0 

2064.17 

134.17 

2198.34 

500 

1903       5     1 

1        3       26 

1698.34 

134.73 

100 

1904       7     5 

1        2         4 

1698.34 

120.02 

1953.09 

800 

1904       8    5 

010 

1053.09 

5.27 

1058.36 

Ans. 

MOGUL'S  COM.  AR.  —  18 


\ 


274  PERCENTAGE   AND   ITS   APPLICATIONS  [§709 

709.  RULE.  Find  the  amount  of  tlw  principal  to  the  time 
when  the  payment  or  the  sum  of  the  payments  shall  exceed 
the  interest  then  due. 

From  this  amount  deduct  tlw  payment  or  the  sum  of  the 
payments  made. 

Consider  the  remainder  as  a  new  principal,  and  proceed 
as  before  to  the  time  of  settlement. 

7-u-M  WRITTEN  EXERCISE 

/?  ^-.7.  On  a  mortgage  for  $650,  made  Aug.  10,  1894,  and  bearing 
i*  interest  at  6%,  payments  were  indorsed  as  follows:  Feb.  2,  1896, 
7*  $100;  June  20,  1898,  $50;  Nov.  1,  1900,  $250.  How  much  was 
T*  |  due  Mar.  31, 1903  ?  .|ft 

On  a  claim  for  $3000,  dated  Sept.  15,  1900,  bearing  interest 
at  8%,  payments  were  made  as  follows :  Jan.  1, 1901,  $300 ;  July  20, 
1901,  $500;  Feb.  2,  1902,  $  125;  Apr.  20,  1903,  $1800.  How  much 
was  due  at  final  settlement  Jan.  1,  1904? 

lr» ""  3.   On  the  note  below  indorsements  were  made  as  follows :  Apr. 
1901,  $300;  Nov.  20,  1902,  $1000;  Mar.  18,  1903,  $600;  Mar. 
'"1904,  $  1100.     What  was  the  balance  due  Jan.  1,  1905  ? 

$4000.00  Chicago,  111.,  Mar.  15,  1901.      v 

On  demand  we  promise  to  pay  to  the  order  of  Williston,  Burgess 

&   Hart,  Four  Thousand  Dollars,  at  Union  National  Bank,  with 

interest  at  6%.. 

Value  received.  HOUGHTON,  DUTTON  &  Co*:  Ji&7.si 

\ 

4.  A  note  of  $  1500,  dated  June  20, 1902,  bearing  interest  at 
had  payments   indorsed  upon  it  as  vfollows :   Dec.  5,  1902, 
Apr.  2,  1903,  $30;  July  20, 1903,  $500;  Dec.  31,  1903,  $400. 
the  amount  due  Apr.  1,  1904. 

5.  On  a  mortgage  for  $4500,  dated  May  1,  1899,  and  bearing  in- 
at   7%,  the  following  payments  were  made:    Feb.  2,  1900, 

Aug.  5,  1900,  $75-,  Aug.  5,  1901,  $2000;   Dec.  20,  190L', 
$300.     How  much  was  due  at  final  settlement  Apr.  1,  1903? 

-J4      „*-'*" 

.JM  -  "  6.   On  the  note  below  payments  were  indorsed  as  follows  :  Oct.  1, 

.  Co 


§§  709-713] 


PARTIAL   PAYMENTS 


275 


/^>* 


1901,  $750;  Apr.  1,  1902,  $150;    Oct.  1,  1902,  $365.90;  Mar.  25, 
1903,  $150;  Nov.  1,  1903,  $200.    How  much  was  due  Apr.  6, 1904? 

$2500.00 


r  7  *-_ 


Boston,  Mass.,  Apr.  6,  1901.      f^,  ^ 

Two  years  after  date  for  value  received,  I  promise  to  pay  toj^— :' 
the  order  of  C.  W.  Frey  &  Co.,  Twenty-five  Hundred  Dollars, 


interest  at  6%. 


F.  M.  EJ.LEBY. 


MERCHANTS'  RULE 


710.  When  partial  payments  are  made  on  interest-bearing 

that  run  for  one  year  or  less,  the  amount  due  at  final  settlement  is 
usually  found  by  the  Merchants'  Rule. 

711.  General  Principles.     The  Merchants'  Eule  recognizes  the 
following  general  principles  : 

1.  The  face  of  the  note  draws  interest  to  the  time  of  settlement. 

2.  Interest  is  allowed  on  each  payment  from  the  time  -such  pay- 
ment is  made  to  the  date  of  settlement. 

The  Merchants'  Rule  is  varied  in  its  use  by  different  creditors,  and  hence  is 
rather  more  an  agreement,  founded  upon  custom  or  otherwise  between  debtor 
and  creditor  as  to  mode  of  settlement,  than  a  strict  rule  of  law. 

712.  To  find  the  balance  due  by  the  Merchants'  Rule  for  partial 
payments. 

713.  Example.     A  note  for  $  900,  dated  May  5,  1903,  payable  on 
demand,  shows  that  the  following  payments  have  been  made  :   June 
20,  $200;  Aug.  15,  $300;  DeC.  1,  $200.     What  is  due  Dec.  31, 
1903,  money  being  worth  6%  ? 

SOLUTION 
Face  of  note          .........  $900.00 

Interest  from  May  5,  1903,  to  Dec.  31,  1903(7  ino.  26  da.)   .  35.40 

Amount  of  note  at  date  of  final  settlement,  Dec.  31,  1903    .  935.40 

First  payment        .........     $200.00 

Interest  on  payment  from  June  20  to  Dec.  31  (6  mo.  11  da.)  6.37 

Second  payment    .........        300.00 

Interest  on  payment  from  Aug.  15  to  Dec.  31  (4  mo.  16  da.)  6.80 

Third  payment      .........        200.00 

Interest  on  payment  from  Dec.  1  to  Dec.  31  (30  da.)  .        .  1.00 

Value  of  payments  at  the  date  of  final  settlement         .         .  714.17 

Balance  due  Dec.  31,  the  date  of  final  settlement          .        .  $221.23 


276 


PERCENTAGE   AND   ITS  APPLICATIONS 
CONDENSED  FORM  FOR  WRITTEN  WORK 


713-714 


DATES 

INTEREST 
PERIODS 

PRINCI- 
PAL 

PAY- 
MENTS 

INTERESTS 

AMOUNT  OF 
PRINCIPAL 

AMOUNTS  OF 
PAYMENTS 

BALANCE 

Mo.         Da. 

Mo.         Da. 

6          5 

7            26 

$900 

$35.40 

$  935.40 

$935.40 

6         20 

6         11 

$200 

6.37 

$206.37 

8         15 

4         16 

300 

6.80 

306.80 

12           1 

0         80 

200 

1.00 

201.00 

714.17 

12         31 

$221.23 

Ans. 

714.   EULE.    Find  the  amount  of  the  principal  to  the  date  of 

settlement  regardless  of  any  payments  made. 

Find  the  amount  of  each  payment  from  the  time  it  was 

made  to  the  time  of  settlement- 
Subtract  the  sum  of  the  payment  amounts  from  the  amount . 

of  the  principal  and  the  result  will  be  the  balance  due. 

WRITTEN  EXERCISE 

1.  A  note  for  $2100  dated  Apr.  15,  1903,  payable  on  demand, 
with  interest,  bears  the  following  indorsements:   June  20,  $300; 
Sept   1,  $200;  Nov.  25,  $750;  Dec.  18,  $300,    What  is  due  Jan. 
31, 1904,  money  being  worth  7%  ? 

2.  What  is  the  balance  due  Apr.  15,  1904,  on  a  note  for  $525 
dated  Jan.  1,  1903,  bearing  6%  interest  if  the  following  indorse- 
ments were  made  thereon:  Mar.  2,  1903,  $75;  July  15,  1903,  $200; 
Sept.  20,  1903,  $100;   Jan.  3,  1904,  $50? 

8.  A  note  for  $  1600  dated  Mar.  2,  1903,  payable  in  6  months 
.with  interest  at  6%  has  the  following  indorsements:  Apr.~3,  $450; 
June  15,  $320;  July  19,  $179.85;  Aug.  3,  $400.  What  is  due  at 
the  maturity  of  the  note  ? 

4.  A  note  for  $950.75  dated  Mar.  8,  1903,  bears  the  following 
indorsements:  Apr.  16,  $250;  June 8,  $150;  Aug.  2,  $200;  Sept.  30, 
$100;  Nov.  2,  $90.  riWhat  was  due  Dec.  8, 1903,  at  6%  ?  At  5%  ? 
At8%?  i-VC\ 

6.  A  note  for  $  1900  dated  Jan.  25, 1903,  was  indorsed  as  follows ; 
Apr.  2,  $900;  May  3,  $750;  July  3,  $100.  What  remained  due 
Sept.  25,  1903,  money  being  worth  G  %  ? 


§§  716-721]  EQUATION  OF  ACCOUNTS  277 

EQUATION  OF  ACCOUNTS 

715.  Equation  of  accounts  is  the  process  of  finding  the  date  when 
the  balance  of  an  account  can  be  paid  without  loss  or  gain  to  either 
party. 

716.  Accounts  having  items  on  but  one  side,  either  debit  or 
credit,  are  called  simple  accounts,  and  the  equating  of  such  accounts  is 
called  simple  equation,  or  equation  of  bills. 

717.  Accounts  having  both  debit  and  credit  items  are  called 
compound  accounts,  and  the  equating  of  such  accounts  is  called  com- 
pound equation,  or  equation  of  accounts. 

718.  The  term  of  credit  is  the  time  that  must  elapse  before  a 
debt  becomes  due. 

If  the  term  of  credit  is  given  in  days,  the  exact  number  of  days  must  be 
added  to  the  date  of  the  purchase  or  sale  ;  if  given  in  months,  the  number  of 
months,  regardless  of  the  number  of  days  included,  must  be  added  to  the  date 
of  the  purchase  or  sale. 

719.  Book  accounts  bear  legal  interest  after  they  become  due ; 
and  notes,  even  if  not  containing  an  interest  clause,  bear  interest 
after  maturity. 

720.  The  average  term  of  credit  is  the  time  that  must  elapse 
before  several  debits  due  at  different  times  may  be  equitably  dis- 
charged in  one  sum. 

721.  The  average  date  of  payment,  the  equated  date,  or  due  date 

is  the  date  on  which  payment  or  settlement  may  be  equitably  made. 
To  illustrate,  suppose  A  stands  charged  as  follows  : 

Jan.  10,  $200. 

Jan.  20,  $200. 

Jan.  30,  -$200. 

If  the  first  charge  were  not  paid  on  Jan.  10,  it  would  be  subject  to  interest 
from  Jan.  10  to  the  date  of  settlement ;  if  the  third  charge  were  paid  before 
Jan.  30,  discount  should  be  allowed  for  the  number  of  days  between  the  date  of 
settlement  and  Jan.  30  ;  if  the  second  charge  were  paid  Jan.  20  no  interest 
would  be  charged  or  discount  allowed.  It  will  "be  seen  that  the  interest  on  the 
first  charge  from  Jan.  10  to  Jan.  20  is  equal  to  the  discount  on  the  third  charge 
from  Jan.. 20  to  Jan.  30.  Hence,  the  whole  account  can  be  equitably  paid  on 
Jan.  20,  the  average  date  of  payment. 


278  PERCENTAGE  AND  ITS  APPLICATIONS         [§§  722-725 

722.  The  focal  date  is  any  assumed  date  of  settlement  with 
which  the  dates  of  several  accounts  are  compared  for  the  purpose 
of  finding  the  average  term  of  credit,  or  due  date. 

Any  date  may  be  used  as  a  focal  date,  and  the  result  will  be  the  same.  This 
is  true  because  all  items  are  equally  affected  when  a  different  focal  date  is  used. 

Any  rate  per  cent  may  be  used  and  the  result  will  be  the  same.  As  a  matter 
of  convenience  always  use  6%,  and  base  all  computations  on  the  commercial 
year  of  360  days. 

723.  Only  personal   accounts    are  equated.     The   occasion  for 
equating  personal  accounts  arises  from  two  causes : 

1.  If  any  item  of  any  account  is  not  paid  when  due,  the  holder 
of  the  account  suffers  a  loss. 

2.  If  an  item  is  paid  before  it  is  due,  the  holder  of  the  account 
realizes  a  gain. 

724.  Accounts  are  equated  to  ascertain  a  date  when  the  settle- 
ment may  be  made  without  loss  or  gain  to  either  the  holder  of  the 
account  or  the  maker  of  it. 

725.  The  face  value  of  an  item  is  always  to  be  used  in  equating 
accounts. 

An  item  not  subject  to  a  term  of  credit  is  worth  its  face  value  the  day  it  is 
dated.  This  is  always  true  of  an  interest-bearing  note. 

Items  subject  to  a  term  of  credit  and  non-interest-bearing  notes  are  worth 
their  face  value  at  maturity. 

DRILL  EXERCISE 

1.  How  long  may  $  2  be  kept  to  balance  the  use  of  $  4  for  10 
days  ?     $  1  for  20  days  ?     $  6  for  15  days  ? 

2.  The  use  of  f  40  for  1  month  is  equivalent  to  the  use  of  what 
sum  for  2  months  ? 

3.  If  I  use  1 6  of  B's  money  for  30  days,  how  much  of  my  money 
should  he  use  for  10  days  in  return  for  the  accommodation  ? 

4.  If  I  pay  one  half  of  an  account  10  days  before  the  whole 
account  is  due,  how  long  after  the  whole  account  is  due  may  I  have 
in  which  to  pay  the  balance  ? 

6.  If  I  pay  $10  of  an  account  30  days  before  it  is  due,  how  long 
may  I  keep  $  5,  the  balance  account,  after  maturity  ?  $  20  ? 


§725]  EQUATION  OF  ACCOUNTS  279 

6.  Jan.  10  Mason  &  Brown  sold  F.  E.  Rogers  on  account  30  days 
merchandise  amounting  to  $400.      (a)  When  is  the  account  due? 
(6)  If  on  Jan.  25  F.  E.  Rogers  paid  $  200,  on  what  date  is  the  balance 
due? 

SOLUTIONS,     (a)  The  account  is  due  30  days  after  Jan.  10,  or  Feb.  9. 

(6)  If  a  payment  of  $200  was  made  on  Jan.  25,  Mason  &  Brown  have  had 
the  use  of  $200  for  15  days ;  hence,  they  should  extend  the  time  on  the  remain- 
ing $200  15  days  beyond  the  original  maturity.  Feb.  9  plus  15  days  equals 
Feb.  24,  the  date  on  which  the  remaining  $  200  of  the  account  should  be  paid. 

7.  May  5  F.  C.  Clark  &  Co.  sold  Charles  H.  Jones  &  Son  on 
account  30  days  merchandise  amounting  to  $  400.     May  20  Charles 
H.  Jones  &  Son  made  a  payment  of  $  100  on  account.     On  what  date 
should  the  balance  be  paid  without  loss  or  gain  ? 

SOLUTION.  The  original  charge  matures  30  days  after  May  5,  or  June  4. 
The  amount  paid,  $100,  is  |  of  the  amount  remaining  unpaid,  $300.  Since 
$100  is  paid  15  days  before  it  is  due,  the  $300  may  be  kept  £  of  15  days,  or  5 
days,  after  it  is  due.  June  4  plus  5  days  equals  June  9,  the  date  on  which  the 
remaining  $  300  should  be  paid  without  loss  or  gain. 

8.  Nov.  1  George  B.  Thayer  &  Co.  sold  E.  M.  Williams  on 
account  30  days  merchandise  amounting  to  $  600.     (a)  When  is  the 
account  due  ?     (6)  If  a  payment  of  $  300  is  made  on  Nov.  1,  on  what 
date  is  the  balance  due  ? 

9.  Nov.  1  W.  D.  Lyman  sold  F.  C.  Hill  on  account  30  days 
merchandise  amounting  to  $  600.     If  a  payment  of  $  200  is  made  on 
Nov.  11,  on  what  date  is  the  balance  due  ?     A  payment  of  $  100  ? 
A  payment  of  $400? 

10.  Mar.  20  W.  K.  Frey  purchased  a  bill  of  goods  of  you  amount- 
ing to  $300.     If  no  term  of  credit  is  given,  how  much  was  legally 
due  Mar.  30  ? 

11.  Smith  &  Brown  bought  goods  of  you  as  follows : 

Mar.  20,  $300. 
Mar.  30,  $300. 

(a)  How  much  is  legally  due  on  the  above  account  Mar.  30  ? 
(6)  On  what  day  can  the  amount  of  the  account,  $600,  be  paid 
without  interest  ? 

SOLUTIONS,  (a)  If  the  account  is  settled  Mar.  30,  on  the  charge  of  Mar.  20 
there  is  10  days'  interest  due.  The  interest  on  $ 300  for  10  days  is  $.50,  which, 
added  to  the  amount  of  the  account,  $  600,  makes  the  amount  legally  due  $  600.50. 


280  PERCENTAGE   AND   ITS  APPLICATIONS         [§§  725-727 

(6)  10  days'  interest  on  $300  is  equivalent  to  5  days'  interest  on  $600; 
therefore  if  the  whole  account  had  been  paid  5  days  before  Mar.  30,  or  Mar.  25, 
only  the  face  of  the  account,  $  600,  would  have  been  due. 

PROOF.  The  interest  on  the  first  charge  from  Mar.  20  to  Mar.  25  is  $.25, 
and  the  discount  on  the  second  charge  from  Mar.  25  to  Mar.  30  is  $.25.  The 
interest  and  discount  being  equal,  the  face  of  the  account,  $600,  can  be  paid 
Mar.  25  without  loss  or  gain  to  either  party. 

12.  If  I  owe  $200  due  May  1,  and  $400  due  May  31,  at  what 
time  can  both  debts  be  equitably  paid  ? 

SIMPLE  EQUATIONS 

726.  To  find  the  equated  time  of  an  account  when  the  items  are  all 
on  one  side  and  are  subject  to  no  terms  of  credit. 

727.  Example.    Robert  S.  Campbell  is  charged  on  the  books  of 
Spencer,  Mead  &  Co.  as  follows : 

Nov.  1,  To  mdse.,  $  60. 
Nov.  7,  To  mdse.,  120. 
Nov.  13,  To  mdse.,  180. 
Nov.  19,  To  mdse.,  240. 

On  what  date  may  the  amount  of  the  account,  $600,  be  paid 

without  interest  ? 

SOLUTION.    For  convenience 

1          <T«n         •£"?  <Tl*  assume  that  the  account  is  being 

8da*  ®'18  settled  Nov.  19.     The  amount  of 

7                           12da"  «24  the  first  charge,  $60,  would  then 

Nov.  13            180            6  da.  .18  draw  interest  from  Nov.  1  to  Nov. 

Nov.  19            240           0  da.  .00  19,  or  for  18  days.    The  interest 

$~600  $  .60  on  $60  for  18  days  equals  $.18. 

,„,  .  ~       The  amount  of  the  second  charge, 
Themtereston$600forlda.=$.10.     ^^  would  draw  interegt  fr*m 

$  .60  -r-  f  .10  =  6,  or  6  da.  Nov.  7  to  Nov.  19,  or  for  12  days. 

Nov.  19-6  da.  =  Nov.  13, 1903,  the     The  ;nterest  on  ®  12°  for  12  **** 

equals  $.24.    The  amount  of  the 

third  charge  would  draw  interest 

from  Nov.  13  to  Nov.  19,  or  for  6  days.  The  interest  on  $  180  for  6  days  equals 
$.18.  The  amount  of  the  fourth  charge,  being  dated  on  the  assumed  date  of 
settlement,  will  not  draw  interest.  Adding  the  several  interest  charges,  the 
amount  is  found  to  be  $.60. 


§§  727-728]  EQUATION  OP   ACCOUNTS  281 

It  will  be  seen  that  if  the  account  were  settled  Nov.  19,  a  payment  of  $  600.60 
would  be  required  ;  but  the  question  is  not,  "  What  is  the  cash  balance  due 
Nov.  19  ?  "  but  "  When  may  the  amount  of  the  account,  $600,  be  paid  without 
interest  ?  "  Hence,  we  have  given  the  principal,  interest,  and  rate  to  find  the 
time  in  days.  The  interest  on  $600  for  1  day  is  $.10.  $.60  is  6  times  $.10. 
Therefore,  there  are  6  days'  interest  due  on  $600  Nov.  19,  and  to  pay  the  amount 
of  the  account  without  interest,  settlement  must  be  made  6  days  before  Nov.  19, 
or  Nov.  13. 

PROOF.  To  prove  that  $  600  may  be  paid  on  Nov.  13  without  interest,  it  is 
necessary  to  show  that  the  interest  on  the  charges  before  Nov.  13  is  equal  to 
the  discounts  to  be  allowed  on  the  charges  made  after  that  date. 

• 

SOLUTION 
The  items  that  fall  due  before  Nov.  13  are  the  following : 

Dates  Items  Interest  Periods  Interests 

Nov.  1  $60  12  da.  $.12 

Nov.  J  120  6  da.  .12 

Total  interest  due  Nov.  13,      $  .24 

The  item  that  falls  due  after  Nov.  13  is  the  following : 

Date  Item  Interest  Period  Interest 

Nov.  19  $240  6  da.  $.24 

The  discount  on  the  item  charged  after  Nov.  13  equals  $  .24. 

Since  the  interest  on  the  items  due  before  Nov.  13  is  equal  to  the  discount 
to  be  allowed  on  the  item  due  after  Nov.  13,  it  is  proved  that  on  Nov.  13  only  the 
face  value  of  the  account  was  due.  Hence  the  equated  date  is  Nov.  13. 

728.  From  the  foregoing  explanations  the  following  rule  may  be 
derived : 

Select  the  latest  date  as  the  focal  date. 

Find  the  exact  time  in  days  from  the  date  of  each  item  to 
the  focal  date  and  compute  the  interest  on^  these  items  for  the 
time  as  found. 

Divide  the  sum  of  the  interest  on  the  several  items  by  the 
interest  on  the  face  of  the  account  for  one  day,  and  the  quotient 
will  be  the  number  of  days  average  time. 

Count  bach  from  the  focal  date  the  number  of  days  so  found 
and  the  date  thus  reached  will  be  the  date  on  which  the  face  of 
the  account  may  be  paid  without  loss  or  gain  to  either  party. 

In  finding  the  average  term  of  credit  in  days,  fractions  of  a  day  of  one 
half  or  greater  are  counted  as  a  full  day ;  fractions  less  than  one  half  day  are 
rejected. 


282 


PERCENTAGE   AND   ITS  APPLICATIONS         [§§  728-730 


Any  date  may  be  taken  as  a  focal  date,  the  only  question  involved  being  a 
balance  of  interest  or  discount,  but  except  for  illustrative  purposes  by  the 
teacher  or  test  exercises  for  advanced  pupils,  the  selection  of  any  date  except 
the  latest  date  for  the  focal  date  is  not  recommended. 

The  selection  of  the  latest  date  as  the  focal  date  removes  the  objection  often 
raised  in  case  an  earlier  date  or  the  earliest  date  be  chosen ;  namely,  that  the 
account  is  not  likely  to  be  settled  before  it  was  made. 


WRITTEN  EXERCISE 

Find  the  equated  date  of  the  following  accounts  of  Lord,  Taylor 
&  Lord.     Prove  the  work  of  each  problem. 


1903 


1.   John  E.  Parker  &  Co.,  Dr.         S.  Patterson,  Gould  &  Co.,  Dr. 

1903 

June    6,  To  mdse.  .  .  $  500. 

June  20,  To  mdse.  .  .  $  200. 

June  29,  To  mdse.  .  .  $  150. 

July    2,  To  mdse.  .  .  $  300. 

July  18,  To  mdse.  .  .  $200. 

4.  T.  L.  King,  Dr. 
1903 

Dec.  1,  To  mdse.  .  .  $450. 
Dec.  15,  To  mdse.  .  .  $  300. 
Dec.  30,  To  mdse.  .  .  $  150. 
1904 

Jan.  8,  To  mdse.  .  .  $  200. 
Jan.  18,  To  mdse.  .  .  $300. 

729.  To  find  the  equated  date  when  the  items  are  all  on  one  side  and 
terms  of  credit  are  given. 

730.  Example.     John  Pierce  is  debited  on  the  books  of  Benedict 
&  Schuier  as  follows : 

1903 

Oct.  1,  To  mdse.,  30  da.,  $600. 
Oct.  8,  To  mdse.,  10  da.,  300. 
Oct.  9,  To  mdse.,  15  da.,  240. 
Oct.  20,  To  mdse.,  10  da.,  360. 


Oct.     5,  To  mdse. 

$  100. 

Oct.     9,  To  mdse.      .    '. 

$150. 

Oct.   16,  To  mdse.      .     . 

$200. 

Oct.   25,  To  mdse.      .     . 

$100. 

Oct.   31,  To  mdse.      .     . 

$200. 

2.   C.  A.  Ruggles,  Dr. 

1903 

Nov.    1,  To  mdse.      .    . 

$150. 

Nov.    8,  To  mdse.      .     . 

$200. 

Nov.  15,  To  mdse.      .    . 

$120. 

Nov.  21,  To  mdse.     .    . 

$150. 

Nov.  28,  To  mdse. 

$200. 

§§  730-731]  EQUATION  OF  ACCOUNTS  283 

When  is  the  face  amount  of  the  account  due  by  equation  ? 

Dates         Terms  of  Credit         Due  Dates         Items          Interest  Periods         Interests 

Oct.  1     30  da.      Oct.  31    $600  00  da.  $.00 

Oct.  8     10  da.      Oct.  18     300  13  da.  .65 

Oct.  9     15  da.      Oct.  24     240  7  da.  .28 

Oct.  20     10  da.      Oct.  30     360  1  da.  .06 

$1500  $.99 

On  Oct.  31,  1903,  $  1500  -f  $  .99,  or  $  1500.99  is  due  on  the  account. 

Since  there  is  interest  due  on  Oct.  31,  1903,  the  face  amount  of  the  account 
was  due  before  Oct.  31,  1903. 

The  interest  on  the  face  amount  of  the  account,  $  1500,  for  1  day  is  $  .25. 

The  total  interests  due  on  the  face  amount  of  the  account  Oct.  31, 1903,  is$  .99. 

$  .99  H-  $  .25  =  4. 

Therefore  it  will  take  $  1500  4  days  to  yield  $  .99  interest. 

4  days'  interest  is  due  Oct.  31,  1903  ;  hence  the  face  amount  of  the  account 
was  due  4  days  before  Oct.  31,  1903. 

Oct.  31, 1903,  minus  4  days  equals  Oct.  27, 1903,  the  equated  date  of  payment. 

PROOF.  To  prove  that  Oct.  27  is  the  equated  date  of  payment,  it  is  necessary 
to  show  that  the  interest  due  on  the  items  before  that  date  is  equal  to  the  dis- 
count to  be  allowed  on  the  items  due  after  that  date. 

SOLUTION 

Dates                     Items                    Interest  Periods  Interests 

Oct.  18               $300                       9  da,  $.46 

Oct.24                 240                       Sda.  .12 

Total  interest,  $.57 

Dates  Items  Discount  Periods  Discounts 

Oct.  30  $360  Sda,  $.18 

•      Oct.81  600  4da.  .40 

Total  discount,      $.68 

In  finding  the  average  term  of  payment  there  was  a  fractional  number  of 
days.  Since  all  fractions  less  than  one  half  day  are  dropped  and  all  fractions 
one  half  day  or  more  are  called  another  whole  day,  if  the  difference  between 
the  interest  and  the  discount  is  less  than  one  half  the  interest  or  discount  on 
the  face  amount  of  the  account  for  1  day,  the  equated  date  used  is  proved  to  be 
correct.  In  the  above  proof  the  difference  between  the  interest  and  discount 
is  $.01,  or  less  than  one  half  the  interest  on  the  amount  of  the  account  for 
1  day ;  hence  Oct.  27,  1903,  is  proved  to  be  the  equated  date  of  payment. 

731.  From  the  foregoing  explanations  the  following  rule  may  be 
derived : 

To  the  date  of  each  item  add  the  term  of  credit  and  proceed  as 
in  728. 


284 


FEKCE^TAGE  AND  ITS  APPLICATIONS         [§§  731-733 


WRITTEN  EXERCISE 


Find  the  equated  date  of  the  following  accounts  of  C.  W.  Allen 
&  Co.     Prove  the  work  of  each  problem. 


1.   Greene  &  Sloan,  Dr. 
1903 

Jan.  2,  To  mdse.,  30  da.,  $200. 
Jan.  28,  To  mdse.,  60  da.,  300. 
Apr.  15,  To  mdse.,  30  da.,  150. 
Apr.  2,  To  mdse.,  60  da.,  240. 


&  Groves  &  Co.,  Dr. 
1903 

Jan.  30,  To  mdse.,  1  mo.,  $  300. 
Feb.  28,  To  mdse.,  60  da.,  300. 
Mar.  25,  To  mdse.,  2  mo.,  1200. 
June  20,  To  mdse.,  30  da.,  1200. 


8.  Gibbons  &  Stone,  Dr.  4.  Steven  Brackett,  Dr 

1903  1903 

May     8,  To  mdse.,  net,     $250.  Oct.    5,  To  mdse.,  30  da.,  $319,50. 

May  25,  To  mdse.,  30  da.,    200.  Oct.  20,  To  mdse.,  4  mo.,    750.00. 

June  18,  To  mdse.,  30  da.,   420.  Dec.   1,  To  mdse.,  2  mo.,    280.50. 

June  30,  To  mdse.,  30  da.,    480.  Dec.  31,  To  mdse.,  1  mo.,    415.90, 


5.  F.  H.  Harper  &  Sons,  Cr. 
1903 

Jan.  20,  To  mdse.,  3  mo.,  $  180. 
Jan.  25,  To  mdse.,  3  mo.,  420. 
Mar.  8,  To  mdse.,  3  mo.,  480. 
Mar.  18,  To  mdse.,  3  mo.,  120. 


6.  F.  H.  Spencer  &  Co.,  Cr 
1903 

Sept.  18,  To  mdse.,  1  mo.,  $1000. 
Sept.  30,  To  mdse.,  60  da.,  200. 
Nov.  8,  To  mdse.,  60  da.,  420. 
Dec.  1,  To  mdse.,  30  da.,  180. 


7.  Shepard  &  Norwell  Co.  sold  C.  W.  Davis  the  following  bills  of 
merchandise :  May  3, 1903,  $2400  on  30  days'  credit;  June  20, 1903, 
$180  on  2  months'  credit;  Aug.  20,  1903,  $1200  on  30  days'  credit; 
Aug.  31,  1903,  $420  on  60  days'. credit.  On  what  date  may  the 
several  bills  be  equitably  paid  in  one  sum  ? 


COMPOUND  EQUATIONS 

732.  To  find  the  equated  date  when  the  account  has  both  debits  and 
credits  and  no  terms  ot  credit  are  given. 

733.  Examples.     7.   Find  the  equated  date  for  paying  the  balance 
of  the  following  account: 


§733] 


EQUATION  OF  ACCOUNTS 


286 


64zx^7  ^^X  Us^J^V^^ 


/7~ 

2 

>  -rev 

^^^ 

^ 

2  r 

^^^ 

3.00 

;  -T 

300 

^ 

J  00 

SOLUTION 
Take  Apr.  II,  1903,  as  the  focal  date. 


Dates 
Mar.    1 
Mar.  16 


Dates 
Mar.  26 
Apr.  11 


Items 

$600 

300 

$900 


Interest  Periods 
41  da. 
27  da. 


Credits 


Interest  Periods 

17  da. 
00  da. 


Interests 
$4.10 
1.35 
$5.45 


Interests 

$.85 

.00 

.85 


$  900  -  $  600  =  $  300,  the  balance  of  the  account. 

$  5.45  -  $  .85  =  $  4.60,  the  interest  due  the  holder  of  the  account  Apr.  11, 1904. 
The  interest  on  $  300  for  1  day  =  $  .05. 
$4.60-7- $.05  =  92. 

Therefore  $300  will  yield  $4.60  interest  in  92  days. 

Apr.  11,  1904,  Jaines  B.  Halsey  not  only  owes  the  balance  of  the  account, 
$  300,  but  92  days'  interest  on  this  amount. 

Therefore  the  face  of  the  account  was  due  92  days  before  Apr.  11,  1904. 
Apr.  11,  1904,  —  92  days  =  Jan.  10,  the  equated  date  of  payment. 

PROOF.  To  prove  the  correctness  of  the  above  work  it  is  necessary  to  show 
that  the  payment  of  $  300  on  Jan.  10,  1904,  will  result  in  neither  a  loss  nor  gain 
to  either  debtor  or  creditor.  This  may  be  done  by  equating  the  account  again 
with  Jan.  10  as  the  focal  date. 

SOLUTION 


Debits 


Dates 
Mar.    1 
Mar.  16 


Items 

$600 

800 


Interest  Periods  Interests 

51  da.  $6.10 
65  da.  3.25 

Total  debit  interest,  $8.35 


286 


PEKCENTAGE   AND   ITS   APPLICATIONS 


[§733 


Credits 


Bates 
Mar.  25 
Apr.  11 


Items 

$300 
300 


Interest  Periods 
75  da. 
92  da. 


Interests 
$3.75 
$4.60 

$8.35 


Total  credit  interest, 

From  the  above  statement  we  see  that  if  the  holder  of  the  account  had,  on  Jan. 
10,  received  cash  for  each  item  charged  in  the  account,  he  would  have  had  the  use 
of  $  GOO  for  51  days  and  $  300  for  65  days,  and  would  have  gained  $  8.35  interest ; 
but  the  first  payment  of  $  300  was  made  Mar.  25,  and  the  second  Apr.  11.  By 
not  receiving  payment  on  Jan.  10  he  would  not  have  the  use  of  $300  for  75  days 
and  92  days;  hence  he  would  have  lost  $8.35  interest.  Since  this  loss  is  can- 
celed by  a  gain  of  $8.35,  it  is  shown  that  the  balance  of  the  account  may  be 
equitably  settled  Jan.  10. 

2.  Find  the  equated  date  for  paying  the  balance  of  the  following 
account : 


£z 

*£Z~6,j~4., 

600 

>C? 

/SLsC^L^j!^ 

.100 

£/) 

1*0 

^ 

tj£ 

.300 

= 

SOLUTION 
Take  May  1,  1904,  as  the  focal  date. 

Debits 


Dates 
Feb.    1 
Feb.  10 


Terms  of  Credit 

90  da. 
60  da. 


Due  Dates 

May    1 
Apr.  10 


Items 

$600 

300 

$900 


Interest  Periods 
00  da. 

21  da. 


Interests 

$    .00 

1.05 

$1.05 


Credits 


Dates 
Feb.  20 
Mar.  30 


Items 

$300 
300 

$600 


Interest  Periods 

71  da. 
32  da. 


Interests 

$3.55 

1.60 

$5.15 


$  900  -  $  600  =  $  300,  the  balance  of  the  account. 

$5.15  -  $  1.05  =  $4.10,  the  interest  due  Ira  B.  Perkins  May  1. 

The  interest  on  $  300  for  1  day  =  $  .05. 

$4.10  -=-$.05  =  82. 


§§  733-735]  EQUATION   OF  ACCOUNTS  287 

Therefore,  $  300  will  yield  $  4.10  interest  in  82  days.  ' 

On  May  1  there  is  82  days'  interest  due  Ira  B.  Perkins. 

Hence,  the  face  of  the  account  is  not  due  until  82  days  after  May  1. 

May  1  plus  82  days  equals  July  22,  the  equated  date  of  payment. 

PROOF 

Debits 

Due  Dates  Items  Interest  Periods  Interests 

May    1  $600  82  da.  $8.20 

Apr.  10  800  103  da.  5.15 

$13.35 

Credits 

Due  Dates  Items  Interest  Periods 

Feb.  20  $300  153  da. 

Mar.  30  300  114  da. 

The  above  statement  shows  that  on  July  22,  1904,  the  loss  suffered  by  the 
holder  of  the  account  is  canceled  by  the  gain  realized.  Therefore,  the  account 
is  proved  to  be  equitably  due  July  22,  1904. 

In  proving  the  equation  of  accounts,  the  equitable  settlement  of  which  is  found 
to  come  at  a  date  within  the  account  or  between  its  extreme  dates,  the  difference 
between  the  interest  and  discount  of  the  debit  items  from  their  respective  dates 
to  the  due  dates  must  be  offset  or  balanced  by  the  difference  between  the  interest 
and  discount  on  the  credit  items  from  their  respective  dates  to  the  due  date, 
within  one  half  cent  of  the  interest  or  discount  on  the  balance  for  one  day. 

734.  General  Principles.     1.   If  the  larger  interest  is  on  the  larger 
side  of  the  account,  it  shows  that  the  holder  of  the  account  has 
suffered  a  loss  because  items  were  not  paid  when  due;  if  on  the 
smaller  side,  it  shows  that  the  holder  of  the  account  has  gained 
because  items  were  paid  before  they  were  due. 

2.  A  loss  is  offset  by  dating  back;  a  gain  is  offset  by  dating 
forward. 

735.  Hence  the  following  rule : 

Find  the  balance  of  the  account  and  also  the  excess  of  inter- 
est from  the  latest  date  as  the  focal  date. 

If  the  balance  of  account  and  excess  of  interest  are  on  the 
same  sidef  date  back ;  if  on  opposite  sides t  date  forward. 


288 


PERCENTAGE  AND   ITS   APPLICATIONS 


[§  735 


ORAL  EXERCISE 

1.  Oct.  1,  Henry  Ball  &  Co.  sold  F.  E.  Gorham  a  bill  of  merchan- 
dise amounting  to  $800.  Terms:  30  days.  Oct.  11,  F.  E.  Gorham 
made  a  payment  of  $400  on  account.  When  is  the  balance,  $400, 
equitably  due  ? 

SOLUTION.  By  the  terms  of  the  contract  the  account  would  mature  Oct.  31. 
Since  a  payment  of  $  400  is  made  20  days  before  maturity,  Henry  Ball  &  Co. 
have  gained  the  use  of  $400  for  20  days.  To  offset  this  gain  they  should  allow 
F.  E.  Gorham  20  days  beyond  the  original  maturity  of  the  account  in  which 
to  pay  the  balance.  Oct.  31  plus  20  days  is  equal  to  Nov.  20,  the  date  on  which 
the  balance  of  the  account  may  be  equitably  paid. 

Find  the  time  for  equitably  paying  the  balance  of  the  following 
accounts.  Terms:  cash. 

Dr.  Or, 

2.   Oct.      1,  $800;  Oct.    11,  $400. 
8.   Apr.  12,  $300;  Apr.   17,  $150. 

4.  July    5,  $900;  July  20,  $300. 

Find  the  time  for  equitably  paying  the  balance  of  the  following 
accounts.  Terms  :  30  da. 

Dr.  Or. 

5.  Oct.     1,  $600;  Oct.      6,  $200. 

6.  May  10,  $400;  May     6,  $  200. 

7.  June  15,  $800;  June  25,  $600, 


WRITTEN  EXERCISE 

Find  the  equated  date  of  payment  of  each  of  the  following 
accounts.     Prove  all  work. 


1. 


E.  M.  ELDRED  &  Co. 


1903 

1903 

Jan. 

20 

To  mdse. 

600 

Feb. 

8 

By  cash 

300 

Feb. 

25 

To  mdse. 

300 

Mar. 

20 

By  cash 

300 

8  735] 


EQUATION   OF   ACCOUNTS 
VICTOR  H.  BROWN  &  Co. 


289 


1903 

1903 

Jan. 

15 

To  mdse. 

600 

Jan. 

25 

By  cash 

1000 

30 

To  mdse. 

300 

Feb. 

15 

By  cash 

200 

Feb. 

8. 

To  mdse. 

600 

20 

To  mdse. 

300 

B.  N.  SHERWOOD  &  SON 


1903 

1903 

Apr. 

8 

To  mdse. 

420 

Apr. 

18 

By  cash 

240 

20 

To  mdse. 

180 

20 

By  cash 

60 

May 

15 

TO  mdse. 

540 

June 

2 

By  cash 

300 

June 

2 

To  mdse. 

60 

W.  I.  PARKER 


1903 

1903 

Aug. 

5 

To  mdse. 

200 

Sept. 

8 

By  cash 

240 

20 

To  mdse.,  2  mo. 

360 

Oct. 

5 

By  60-da.  note 

240 

Sept. 

15 

To  mdse.,  30  da. 

360 

(no  interest) 

REED  &  HAMLIN 


1903 

1903 

June 

20 

To  mdse.,  30  da. 

300 

July 

1 

By  cash 

100 

30 

To  mdse.,  60  da. 

180 

Aug. 

1 

By  cash 

100 

Aug. 

1 

To  mdse.,  30  da. 

480 

Sept. 

1 

By  cash 

100 

Sept. 

20 

To  mdse.,  30  da. 

120 

Oct. 

1 

By  cash 

100 

1904 

Jan. 

1 

By  cash 

100 

NOTE.     Interest  may  be  computed  on  one  of  the  five  similar  credit  items 
for  the  aggregate  number  of  days. 
MOORE'S  COM.  AR.  — 19 


290 


PERCENTAGE   AND   ITS  APPLICATIONS 


736-738 


EQUATION  OF  ACCOUNTS  SALES 

736.  An  account  sales  is  equated  in  practically  the  same  man- 
ner as  an  ordinary  ledger  account.     The  agent's  charges  constitute 
the  debits  of  the  account,  and  the  gross  sales  the  credits. 

The  agent's  charges  include  freight,  cartage,  storage,  commission,  insurance, 
advertising,  guaranty,  etc. 

737.  When  equating  accounts  sales,  agents  generally  consider 
such  charges  as  freight,  cartage,  storage,  and  insurance,  as  not  due 
until  they  have  been  paid. 

738.  When  goods  are  sold  promptly,  agents  usually  consider 
commission  and  guaranty  as  due  on  the  date  of  the  last  sale.    When, 
the  sales  are  large  and  there  are  long  intervals  between  them,  the 
commission  or  guaranty  is  considered  due  on  the  average  due  date 
of  the  sales. 

When  goods  are  sold  for  cash,  or  on  short  time,  the  account  sales  is  seldom 
averaged. 

WRITTEN  EXERCISE 

1.   Find  when  the  net  proceeds  of  the  following  account  sales  are 
due  by  equation.     Consider  the  commission  as  due  on  the  date  of  the 

BOSTON,  MASS.,  Oct.  8,  1903. 
PARKER,  MONTGOMERY  &  Co. 
Sold  for  the  account  of  'W.  D.  SPRAGUE, 

Buffalo,  KY. 


1903 

Sales 

Sept. 

23 

95  bbl.                               6.<»,  cash 

Oct. 

1 

200  bbl.                                6.™,  1  mo. 

18 

65  bbl.                                 5.80,  60  da. 

Nov. 

3 

110  bbl.                                6.80,  30  da. 

25 

130  bbl.                               6.75,  cash 

Charges 

Sept. 

24 

Freight 

62 

60 

20 

Cartage 

30 

Oct. 

28 

Cash  advanced 

2000 

Nov. 

15 

Cooperage 

5 

25 

Commission,  4% 

§§  738-742]  CASH  BALANCE  291 

2.  Using  the  foregoing  form  for  a  model,  arrange  the  following 
narrative  in  the  form  of  an  account  sales  and  find  when  the  net  pro- 
ceeds are  due  by  equation.  Consider  the  commission  as  due  on  the 
average  due  date  of  the  sales. 

R.  J.  Briggs  &  Co.,  Boston,  Mass.,  sold  for  the  account  and  risk 
of  B.  Sornmers  &  Co.,  Chicago,  111.,  1000  bbl.  potatoes  as  follows : 
Nov.  2, 1903,  400  bbl.  peach  blows  at  $  3,  cash ;  Dec.  1,  1903,  300  bbl. 
pink  eyes  at  $3.50,  30  da.;  Jan.  1,  1904,  100  bbl.  peach  blows  at 
$  3.60,  cash ;  Jan.  25, 1904,  200  bbl.  pink  eyes  at  $  3.50,  30  da.  The 
charges  were  as  follows:  Nov.  1,  1903,  freight,  $  350;  Nov.  1,  1903, 
cartage,  $50;  Nov.  1,  1903,  insurance  and  advertising,  $100; 
commission  and  guaranty,  3%. 


CASH  BALANCE 

739.  Cash  Balance  treats  of  showing  the  balance  or  amount  due 
on  an  account  at  any  given  date. 

740.  The  cash  balance  of  an  account  on  which  interest  is  not 
charged  is  the  difference  between  the  two  sides  of  the  account  in 
the  ledger.     The  cash  balance  of  an  account  on  which  interest  is 
charged  is  the  difference  between  the  two  sides  of  the  account  after 
interest  has  been  added  to  the  items  past  due,  or  deducted  from  the 
items  not  due  at  the  date  of  settlement 

741.  Each  item  of  an  account  equitably  draws  interest  from  the 
time  it  becomes  due  to  the  date  of  settlement,  and  each  item  paid 
before  maturity  is  equitably  entitled  to  discount  for  the  time  from 
the  date  of  payment  to  the  date  it  is  due. 

Whether  interest  is  charged  on  the  items  of  a  running  account  or  not  is 
usually  regulated  by  the  custom  of  the  business,  or  an  agreement  between  the 
parties  thereto.  As  a  rule,  retailers  do  not  charge  interest  on  the  items  of  run- 
ning accounts,  but  frequently  the  balance  of  a  closed  account  is  considered  in- 
terest-bearing from  the  date  the  balance  is  brought  down.  Wholesale  dealers 
usually  charge  interest  on  the  items  of  an  account  at  the  expiration  of  the  time 
specified  in  the  terms  of  credit. 

742.  Example.     When  money  is  worth  6%  per  annum,  what  is 
the  cash  balance  due  on  the  following  account  June  23,  1903  ? 


292 


PERCENTAGE   AND   ITS   APPLICATIONS         [§§  742-743 


30/9 


300 


SOLUTION 


Dates 
Jan.    1 
Jan.  31 


Terms  of  Credit 

Due  Dates                 Items          Interest  Periods 

1  mo. 

Feb. 

1            $  600 

142  da. 

10  da. 

Feb. 

10               1800 

133  da. 

$2400 

Credits 

Dates 

Items 

Interest  Periods 

Interests 

i^b.  19 

$300 

124  da. 

$6.20 

?eb.  28 

300 

115  da. 

5.75 

tor.    6 

300 

109  da. 

5.45 

Interests 

$  14.20 

39.90 

$54.10 


$900 


$  17.40 


$2400  +  $54.10  =  $2454.10,  the  amount  due  on  account  June  23,  1903,  had 
no  payments  been  made. 

$900  +  $  17.40  =  $917.40,  the  value  of  the  payments  on  June  23,  1903. 

$2454.10  -  $917.40  =  S  1536.70,  the  cash  balance  of  the  account  June  23, 
1903. 

743.  From  the  foregoing  explanation  the  following  rule  may  be 
derived : 

Find  the  maturity  of  each  item  of  the  account. 

Compute  the  interest  on  each  item  from  the  date  it  becomes 
due  to  the  date  of  settlement. 

To  the  sum  of  the  debit  items  add  the  sum  of  the  debit  inter- 
ests ;  also  to  the  sum  of  the  credit  items  add  the  sum  of  the  credit 
interests. 

Subtract  these  totals  and  the  result  is  the  cash  balance. 


§743] 


CASH    BALANCE 


293 


WRITTEN   EXERCISE 


/.    If  money  be  worth  7%  per  annum,  what  is  the  cash  balance 
due  on  the  following  account,  July  1,  1903  ? 

HENRY   HARRISON   &  Co. 


1903 

1903 

Jan. 

31 

To  mdse. 

450 

Jan. 

2 

By  mdse. 

600 

Mar. 

30 

To  mdse. 

450 

Feb. 

13 

By  cash 

300 

Mar. 

29 

By  mdse. 

300 

2.   What  is  the  cash  balance  of  the  following  account,  Apr.  1, 
1903,  if  the  money  be  worth  8  %  per  annum  ? 

BENJAMIN  TRACY  &  SON 


1902 

• 

1902 

Aug. 

4 

To  mdse.,  1  mo. 

200 

Oct. 

1 

By  cash 

150 

Sept. 

1 

To  mdse.,  2  mo. 

400 

Nov. 

1 

By  cash 

150 

Oct. 

31 

To  mdse.,  4  mo. 

600 

Dec. 

1 

By  cash 

150 

Dec. 

3 

To  mdse. 

300 

1903 

Jan. 

1 

By  cash 

150 

Feb. 

1 

By  cash 

150 

Mar. 

1 

By  cash 

150 

3.  Equate  the  following  account  and  find  the  cash  balance  due 
Apr.  1,  1903,  if  money  be  worth  7%  per  annum.     Prove  the  work. 

BROWN,  SHIPLEY  &  Co. 


1902 

1903 

Sept. 

9 

To  mdse. 

600 

Jan. 

2 

By  cash 

500 

Oct. 

1 

To  mdse.,  2  mo. 

300 

Mar. 

16 

By  2-mo.  note 

Dec. 

13 

To  mdse.,  1  mo. 

150 

(on  interest) 

100 

1903 

Apr. 

30 

By  3-mo.  note 

Jan. 

31 

To  mdse.  ,  1  mo. 

450 

(no  interest) 

300 

May 

1 

By  cash 

200 

NOTE.     To  find  the  cash  balance  of  an  equated  account. 

Find  the  difference  between  the  equated  date  of  payment  and  the  date  of  settle- 
ment, and  compute  the  interest  on  the  balance  of  the  account  for  this  time.  The 
sum  of  the  interest  thus  found  and  the  balance  of  the  account  is  the  cash  balance 
due  at  the  date  of  settlement. 


29-4 


PERCENTAGE   AND   ITS  APPLICATIONS         [§§  744-746 


BANKERS'   CASH  BALANCE 

744.  Many  bankers   balance   their   accounts  with  their   corre- 
spondents at  regular  intervals,  —  monthly,  quarterly,  semiannually, 
or  yearly,  allow  interest  on  all  sums  that  have  been  credited,  charge 
interest  on  all  sums  that  have  been  debited,  and  bring  the  cash  bal- 
ance down  to  a  new  account  to  subsequently  draw  interest  the  same 
as  the  regular  items  in  the  account. 

745.  Some  bankers  and  trust  companies  balance  their  accounts 
with  depositors  at  regular  intervals  and  allow  interest  on  the  bal- 
ances credited. 

746.  Example.     Find  the  balance  due  on  the  following  account 
Apr.  1,  1904,  settlements  being  made  quarterly  with  interest  at  6%. 


ffff 


£2. 


200 


BA:TK  ACCOUNT  CURRENT 


DATES 

DEBITS 

CREDITS 

CREDIT 
BALANCES 

DAYS 

CREDIT 

INTERESTS 

1904 

Jan. 

1 

800 

800 

3 

40 

4 

100 

700 

4 

47 

8 

1000 

1700 

3 

85 

11 

600 

1200 

30 

6 

Feb. 

10 

800 

2000 

10 

3 

33 

20 

150 

1850 

14 

4 

32 

Mar. 

5 

200 

2050 

25 

8 

54 

30 

50 

2000 

2 

67 

800 

2800 

24 

58 

800 

2000  + 

24 

.68  =  2024 

.58 

§§  746-747] 


CASH   BALANCE 


295 


SOLUTION.  By  arranging  the  debit  and  credit  items  in  the  order  of  their 
dates  the  balance  of  the  account  at  each  of  the  dates  may  easily  be  determined. 
The  account  shows  a  credit  balance  of  $  800  from  Jan.  1  to  Jan.  4,  or  for  3  days, 
when  the  amount  is  reduced  to  $  700  by  the  charge  of  $  100.  The  interest  on  $  800 
for  3  days  is  $  .40.  The  account  shows  a  credit  balance  of  $  700  from  Jan.  4  to 
Jan.  8,  or  for  4  days,  when  the  amount  is  increased  to  $1700  by  the  credit  of 
$1000.  The  interest  on  $700  for  4  days  is  $.47.  Continuing  in  this  manner 
to  Apr.  1,  it  is  found  that  on  that  date  the  account  shows  a  credit  balance  of 
$2000,  and  that  on  the  daily  balances  there  has  accumulated  $24.58  interest. 
The  cash  balance  of  the  account  is  then  found  to  be  $2000  plus  $24.58,  or 
$2024.58. 

NOTE.  Had  there  been  a  debit  balance  on  any  of  the  above  dates,  two  extra 
columns  would  have  been  required  in  the  operation,  —  one  for  the  debit  balances 
and  one  for  the  debit  interests.  The  difference  between  the  debit  and  credit 
interests  would  then  be  the  balance  of  accrued  interest. 

747.   Hence  the  following  rule  may  be  derived : 

Arrange  the  debits  and  credits  in  the  order  of  their  dates 
and  find  the  'balance  of  the  account  at  each  date. 

Find  the  interest  on  each  balance  for  the  period  that  it 
remains  unchanged. 

If  the  balance  of  interest  and  the  balance  of  the  account 
are  on  tJie  same  side,  take  their  sum;  if  on  the  opposite  sides, 
take  their  difference. 

The  result  obtained  is  the  cash  balance  due. 


WRITTEN  EXERCISE 


1.  Find  the  balance  due  on  the  following  bank  account  July  1, 
1903,  at  4%. 

Dr.        CENTRAL  NATIONAL  BANK,  Springfield,  Mass.        Cr. 


1903 

1903 

Apr. 

15 

To  cash 

200 

Apr. 

1 

By  cash 

1200 

20 

To  cash 

200 

May 

4 

By  cash 

900 

May 

20 

To  cash 

300 

June 

1 

By  cash 

500 

June 

10 

To  cash 

300 

20 

By  cash 

420 

#.   Find  the  balance  due  on  the  following  bank  account,  Apr.  1, 
1903,  at  3%. 


296 


PERCENTAGE   AND   ITS  APPLICATIONS 


747-753 


Dr.         MERCHANTS  NATIONAL  BANK,  Kochester,  N.Y.        Cr. 


1903 

1903 

Jan. 

2 

To  cash 

600 

Jan. 

1 

By  cash 

1500 

Feb. 

8 

To  cash 

480 

31 

By  cash 

1200 

21 

To  cash 

240 

Feb. 

15 

By  cash 

120 

Mar. 

20 

To  cash 

180 

Mar. 

31 

By  cash 

400 

8.  The  Security  Trust  Company,  Rochester,  N.Y.,  allows  inter- 
est to  its  depositors  on  daily  balances  at  4%  per  annum,  payable 
quarterly.  Find  the  cash  balance  of  the  following  account  with 
George  W.  Snyder,  Apr.  1,  1903:  Jan.  1,  1903,  deposited  $900; 
Jan.  8,  drew  out  $200;  Jan.  12,  deposited  $750;  Jan.  15,  drew  out 
$475;  Feb.  9,  deposited  $721.90;  Feb.  24,  drew  out  $121.90;  Mar. 
15,  deposited  $795.98  ;  Mar.  30,  drew  out  $400. 

SAVINGS-BANK  ACCOUNTS 

748.  A  savings  bank,  as  its  name  implies,  is  an  institution  organ- 
ized for  the  purpose  of  encouraging  economy  and  thrift  and  caring 
for  the  savings  of  the  people. 

749.  The  deposits  in  savings  banks  are  practically  payable  on 
demand. 

Savings  banks  generally  reserve  the  right  to  require  depositors  to  notify  them 
from  30  to  60  days  before  making  a  withdrawal. 

750.  The  interest  term  is  the  time  between  dates  at  which  divi- 
dends of  interest  are  declared. 

Dividends  of  interest  are  usually  declared  semiannually. 

751.  If  interest  is  not  withdrawn,  it  is  placed  to  the   credit 
of  the  depositor  on  the  books  of  the  bank,  and  draws  interest  the 
same  as  any  regular  deposit.     In  this  way  savings  banks  pay  their 
depositors  compound  interest. 

No  interest  is  allowed  on  fractional  parts  of  a  dollar. 

752.  The  interest  days  are  the  days  on  which  interest  is  allowed 
to  commence. 

753.  Savings  banks  are  not  uniform  in  their  practice  of  allow- 
ing interest  on  deposits  made  after  the  beginning  of  the  interest 
term.     In  some  savings  banks  deposits  begin  to  draw  from  the  first 


§§  753-757] 


SAVINGS-BANK   ACCOUNTS 


297 


of  each  quarter ;  in  others,  from  the  first  of  each  month.    The  latter 
method  is  preferable  for  persons  having  a  small  income. 

Monthly  interest  days  usually  begin  on  the  first  day  of  each  month ;  quar- 
terly interest  days  on  Jan.  1,  Apr.  1,  July  1,  and  Oct.  1 ;  semiannual  interest 
days  on  Jan.  1  and  July  1. 

754.  Nearly  all  savings  banks  allow  interest  on  only  those  sums 
that  have  been  on  interest  for  the  full  time  between  the  interest  days. 

Thus,  if  the  interest  begins  quarterly,  only  those  sums  that  have  been  on 
deposit  for  the  full  quarter  draw  interest ;  if  monthly,  only  those  sums  that  have 
been  on  deposit  for  the  full  month  draw  interest. 

755.  Savings  banks  furnish  each  depositor  with  a  small  book 
called  a  pass  book,  in  which  are  entered  all  amounts  deposited  and 
all  amounts  withdrawn,  together  with  the  interest  credited  to  the 
depositor  at  the  expiration  of  the  interest  term. 

756.  To  find  the  balance  due  a  depositor  when  there  are  no  with- 
drawals. 

757.  Example.     The  interest  term  of  Wildey  Savings  Bank  is 
6  months.     A  deposited  in  this  bank  Dec.  20,  1903,  $200;  Feb.  10, 
1904,  $100;   Apr.  1,  1904,   $50;   June  8,  1904,  $50.     No  with- 
drawals having  been  made,  what  was  due  July  1,  1904,  if  interest 
at  4%  per  annum  be  reckoned  on  the  deposits  (a)  from  the  first  of 
each  quarter  ?  (6)  from  the  first  of  each  month  ? 

(a) 


DATES 

DEPOSITS 

DAILY 
BALANCES 

INTEREST  DAYS 

SMALLEST 
QUARTERLY 
BALANCES 

QUARTERLY 
INTERESTS 

1903 

Dec.  20 

200 

200 

Jan.  1 

1904 

Feb.  10 

100 

300 

Apr.  1 

50 

350 

Apr.  1 

200 

2.00 

June  8 

60 

400 

July  1 

400 

Julyl 

360 

3.50 

6.50 

400. 

405.50 

SOLUTION.    If  interest  begins  on  the  first  of  each  quarter,  only  the  smallest 
balance  for  any  quarter  will  draw  interest.     The  deposit  of  Feb.  10,  being  made 


298 


PERCENTAGE   AND   ITS   APPLICATIONS 


[§757 


after  the  beginning  of  the  first  quarter,  will  not  begin  to  draw  interest  until  the 
beginning  of  the  second  quarter.  Hence,  the  only  sum  that  draws  interest  for 
the  first  quarter  is  the  deposit  of  Dec.  20,  $200.  The  interest  on  $200  for  one 
quarter  is  $2.  The  deposits  of  Feb.  10  and  Apr.  1,  together  with  the  smallest 
balance  for  the  first  quarter,  will  draw  interest  for  the  second  quarter.  The 
deposit  of  June  8,  being  made  after  the  beginning  of  the  second  quarter,  will  not 
draw  interest  until  the  beginning  of  the  third  quarter.  Hence,  the  sum  to  draw 
interest  for  the  second  quarter  is  $350  ($200  +  $  100  +  $50).  The  interest  on 
$350  for  one  quarter  is  $3.50.  $2  -f  $3.50  =  $5.50,  the  interest  to  be  added  to 
the  account  July  1.  $400  +  $5.50  =  $405.50,  the  balance  due  on  the  account 
July  1. 

(W 


DATES 

DEPOSITS 

INTEREST  DATS 

SMALLEST  MONTHLY 
BALANCES 

MONTHLY 
INTERESTS 

1903 

Dec.  20 

200 

Jan.  1 

200 

1904 

Feb.  10 

100 

Feb.  1 

200 

1.00 

Apr.  1 
June  8 

50 
50 

Mar.  1 
Apr.  1 
May  1 
June  1 
July  1 

300 
350 
350 
350 
400 

1.50 
1.75 
1.75 
1.75 

3)7.75 
2.58 

5.17 

400. 

405.17 

SOLUTION.  Only  the  smallest  balance  on  deposit  each  month  will  draw 
interest.  The  amount  deposited  Dec.  20  will  not  begin  to  draw  interest  until 
Jan.  1.  The  smallest  balance  for  each  month  is  as  shown  above.  The  aggregate 
interest  on  the  smallest  monthly  balance  is  found  to  be  $7.75  at  6%,  or  $5.17 
at  4%.  $400,  the  balance  on  deposit  July  1,  +  $5.17  =  $405.17,  the  balance 
due  the  depositor  July  1,  1904. 


WRITTEN  EXERCISE 

1.  A  made  the  following  deposits  in  a  savings  bank :  July  1, 
1903,  $50;  July  30,  $50;  Aug.  20,  $100;  Oct.  5,  $200;  Nov.  8, 
$  150 ;  Dec.  15,  $  200.  If  the  interest  term  is  6  months,  what  is  the 
balance  due  A  Jan.  1,  1904,  interest  being  allowed  on  balances  from 
the  first  day  of  each  quarter  at  4%  per  annum  ? 


§§  757-759] 


SAVINGS-BANK  ACCOUNTS 


299 


2.  J.  M.  Carroll  made  the  following  deposits  in  the  Security 
Savings  Bank:  Dec.  18,  1903,  $400;  Jan.  5,  1904,  $200;  Mar.  8, 
1904,  $  100;  May  20, 1904,  $  30;  July  1,  1904,  $40.  If  the  interest 
term  is  3  months,  what  is  the  balance  due  J.  M.  Carroll  July  1, 
1904,  interest  at  4%  being  computed  from  the  first  day  of  each 
quarter  ? 

758.  To  find  the  amount  due  depositors  when  there  are  withdrawals. 

759.  Example.   Find  the  balance  due  July  1, 1904,  on  the  follow- 
ing account.     Deposits:  Dec.  10,  1903,  $600;  Apr.  10,  1904,  $200; 
May  20,  $150.     Withdrawals:  Mar.  10,  $300;  May  1,  $50.     The 
interest  term  is  6  months  and  interest  at  the  rate  of  4%  per  annum 
is  allowed  from  the  first  day  of  each  quarter. 


DATES 

DEPOSITS 

WITHDRAWALS 

DAILY 

BALANCES 

INTEREST 
DATS 

SMALLEST 
QUARTERLY 
BALANCES 

QUARTERLY 
INTERESTS 

1903 

Dec.  10 

600 

600 

Jan.  1 

1904 

Mar.  10 

300 

300 

Apr.  1 

300 

Apr.l 

300 

3.00 

Apr.  10 

200 

500 

Mayl 

50 

450 

May  20 

150 

600 

July  1 

300 

3.00 

6.00 

600. 

606.00 

SOLUTION.  Interest  begins  quarterly  and  the  interest  days  are  Jan.  1,  Apr.  1, 
and  July  1.  The  smallest  balance  for  the  first  quarter  is  $300,  and  the  smallest 
balance  for  the  second  quarter  is  $  300.  The  quarterly  interest  on  these  -two 
balances  aggregates  $6.00.  $600,  the  amount  on  deposit  July  1,  plus  $6.00, 
the  interest  due  on  that  date,  equals  $  606.00,  the  balance  of  the  account  July  1, 
1904. 

WRITTEN  EXERCISE 

1.  W.  E.  Small  deposits  in  a  savings  bank  as  follows :  Jan.  1, 
$  400 ;  Feb.  2,  $  200 ;  Mar.  10,  $  150 ;  Apr.  2,  $  60 ;  May  18,  $  200 ; 
during  the  same  time  he  withdrew  as  follows:  Jan.  10,  $50;  Feb. 
4,  $  50 ;  Apr.  5,  $  50 ;  June  30,  $  80.  The  interest  term  is  6  months. 
What  interest  at  4%  per  annum,  to  commence  from  the  first  of  each 
quarter,  should  be  added  to  the  account  July  1  ? 


300 


PERCENTAGE   AND   ITS   APPLICATIONS         [§§  759-764 


2.  In  the  Home  Savings  Bank  the  interest  term  is  6  months, 
and  the  interest  days  are  Jan.  1,  Apr.  1,  July  1,  and  Oct.  1.  Find 
the  balance  due  on  the  following  account  J  uly  1,  1904,  at  4  %  per 
annum. 


£00 


JJ2. 


STOCKS 

760.  A  joint  stock  company  is  a  partnership  in  which  the  affairs 
of  the  business  are  conducted  by  officers  chosen  by  the  stockholders. 

761.  A  corporation  is  a  fictitious  person.     It  consists  of  several 
natural  persons  who,  in  the  name  .of  the  corporation,  are  authorized 
by  law  to  transact  business. 

The  instrument  which  defines  the  rights  and  duties  of  the  corporation  is 
called  a  charter.    It  is  issued  by  government  under  seal. 

762.  Stocks  is  a  general  term  applied  to  shares  in  the  capital 
stock  of  banks,  insurance,  railroad,  and  other  incorporated  or  joint 
stock  companies. 

763.  A  stock  certificate  is  a  written  or  printed  instrument  of  a 
joint  stock  company  or  corporation  issued  to  the  stockholder,  certi- 
fying the  value  of  each  share  and  the  number  of  shares  such  cer- 
tificate represents. 

764.  A   share  represents   simply  a  certain  component  part  of 
the  capital  stock.     It  is  commonly  $25,  $50,  or  $100.     The  stock 
certificate  represents  the  number  of  shares  specified  thereon. 


§§  765-774]  STOCKS  301 

765.  The  capital  stock  of  a  company  is  the  sum  of  all  the  shares 
issued  at  their  par  value. 

766.  The  common  stock  of  a  corporation  is  the  stock  which  is 
ordinarily  issued  to  the  incorporators. 

767.  Preferred  stock  is  stock  on  which  dividends  are  paid  before 
any  allowance  is  made  for  dividends  on  the  common  stock. 

Preferred  stock  is  sometimes  issued  to  take  up  the  floating  indebtedness  of 
a  corporation.  Agreed  dividends  are  declared  on  it  at  certain  intervals  out 
of  the  net  earnings,  and  before  any  dividend  can  be  declared  on  the  common 
stock.  Such  stock  is  frequently  issued  upon  the  reorganization  of  railroads  or 
the  consolidation  of  joint  stock  companies. 

768.  The  par  value  of  stocks  is  their  face  value ;  their  market 
value  is  the  sum  at  which  they  are  quoted  in  the  market. 

769.  Stocks  are  above  par  or  at  a  premium  when  they  are  worth 
more  than  their  face  value  ;  below  par  or  at  a  discount  when  they  are 
worth  less  than  their  face  value. 

770.  Stock  quotations  are  published  prices  or  rates  per  share  that 
stocks  sell  for. 

Thus,  when  stock  is  3%  above  par  it  is  quoted  at  103  ;  when  it  is  2  %  below 
par  it  is  quoted  at  98. 

771.  A  dividend  is  a  pro  rata  division  of  profits  among  the  stock- 
holders of  a  company  or  corporation. 

The  income  from  stocks  is  in  the  nature  of  dividends,  and  is  dependent  upon 
the  prosperity  of  the  company  or  corporation.  Dividends  are  declared  at  a 
certain  per  cent  on  the  par  value  of  the  capital  stock  of  the  company,  either 
quarterly,  semiannually,  or  annually.  The  dividend  on  preferred  stock  is  often 
at  a  different  rate  from  that  on  common  stock. 

772.  An  assessment  is  a  sum  levied  pro  rata  upon  the  stock- 
holders of  a  corporation  to  cover  losses,  etc. 

773.  Stock  brokers  are  persons  who  act  for  others  in  buying  and 
selling  stocks  at  a  stock  exchange.     For  this  service  they  charge  a 
certain  rate  per  cent  commission,  called  brokerage,  on  the  par  value 
of  the  stocks  dealt  in. 

774.  Brokerage  is  usually  \%  of  the  par  value  of  the  stock  dealt 
in.     Occasionally  il  is  as  high  as  \  %  or  |  %,  or  as  low  as  y1^  %. 


302  PERCENTAGE   AND   ITS   APPLICATIONS         [§§775-781 

775.  Stocks  are  generally  bought  and  sold  eitker  "  regular  way," 
or  "cash,"  or  "buyer  three,"  or  "seller  three."     Stock  soM  "regular 
way "  is  to  be  paid  for  and  delivered  the  next  day ;    stock  sold 
"  cash  "  is  deliverable  the  day  sold.     When  stock  is  bought  "  seller 
three,"  the  seller  of  the  stock  may  deliver  it  on  any  one  of  the  three 
days  following  the  transaction,  at  his  option,  but  cannot  be  required 
to  deliver  it  till  the  third  day.     When  stock  is  sold  "  buyer  three," 
the  buyer  may  demand  delivery  at  any  time  within  three  days,  but 
is  obliged  to  take  and  pay  for  it  by  the  third  day.     If  a  stock  pays 
a  dividend  while  a  transaction  is   being   executed,   the   dividend 
belongs  to  the  purchaser  of  the  stock. 

776.  A  margin  is  a  deposit  made  with  a  broker  by  a  person  who 
wishes  to  speculate  in  stocks,  such  deposit  being  used  by  the  broker 
to  protect  himself  against  loss.      The  margin  is  usually  10%   of 
the  par  value  of  the  stock  dealt  in. 

A  wishes  to  speculate,  and  deposits  with  B,  his  broker,  $1000  as  a  margin, 
directing  B  to  buy  100  shares  of  a  stock  quoted  at  90.  B  would  pay  for  the  stock 
$9000,  $1000  of  which  is  the  margin  furnished  by  A ;  B  furnishes  $8000,  and 
charges  the  usual  rate  of  interest  on  that  sum  for  "  carrying  "  the  stock.  In  case 
the  quoted  value  of  the  stock  drops  below  90,  the  margin  must  be  made  good  by 
A's  depositing  an  additional  amount.  If  A  fails  to  make  good  his  margin,  B 
may  sell  the  stock  to  protect  himself  from  losing  any  of  the  money  he  has  fur- 
nished. 

777.  Collateral  consists  of  stocks,  notes,  etc.,  given  in  pledge  as 
security  when  money  is  borrowed. 

778.  If  any  one  has  sold  stock  he  does  not  own,  in  the  hope  of 
realizing  a  profit  by  buying  it  in  at  a  lower  price,  he  is  said  to  be 
"short." 

779.  If  stock  has  been  sold  "  short,"  and  the  seller  buys  it  in 
to  realize  a  profit  or  to  protect  himself  against  loss,  he  is  said  to 
"  cover  his  short  sales." 

780.  Stock   sold  Ex.  Div.  means  that  a  recently  declared  divi- 
dend is  received  by  the  seller. 

781.  When  a  corporation  increases  the  quantity  of  its  stock  with- 
out increasing  the  value  of  its  property,  which  the  stock  is  supposed 
to  represent,  the  stock  of  such  a  corporation  is  said  to  be  watered  to 
the  extent  of  the  increase. 


§782] 


STOCKS 


303 


782.  The  following  list,  showing  the  highest,  lowest,  and  closing 
quotations  of  certain  stocks,  and  net  changes  from  closing  prices  of 
the  previous  day,  is  reproduced  from  the  Wall  Street  edition  of  the 
New  York  Sun  under  date  of  Jan  17,  1907. 


High- 

Low- 

THE STOCK  MARKET 

Clos-  Net                            High- 

Low- 

Clos- Net 

est 

est 

ing    CVge 

est 

est 

ing    (. 

W<j 

Allis  Chalm 

15 

15 

15  - 

1 

Great  Nor  pf 

179* 

178 

179 

Amal  Cop 

117* 

115* 

1151- 

t 

Hock  Val  pf 

91* 

91* 

91* 

Am  Beet  Su 

21f 

21* 

21*+ 

i 

Inter-  Met 

36* 

86* 

36*+ 

1 

Am  Car  &  F 

44* 

431 

44  - 

* 

Inter-Met  p 

73* 

73 

73 

Am  Ice  Sees 

86 

85 

86  - 

1 

Iowa  Cent    • 

27 

26* 

27  - 

* 

Am  Loco 

73* 

*  72| 

72*- 

i 

Iowa  Cen  pf 

48* 

47 

48*- 

1* 

Am  Loco  pf 

111* 

111* 

111* 

Kan  C  So  pf 

60* 

60* 

00*- 

f 

Am  Smelt 

151 

150 

150* 

Lou  &  Nash 

142* 

141f 

142*  + 

* 

Am  Sugar 

133* 

132f 

132*- 

f 

Manhattan 

143 

142£ 

143 

Am  Tob  pf 

97 

97 

97 

Mex  Cent 

26 

25* 

26*- 

1 

Anaconda 

282| 

^80 

281*+ 

1* 

M  &  St  L  pf 

89 

89 

89  + 

1 

A  T  &  S  F 

106* 

105* 

1051- 

1 

MStP&SSM 

130* 

130 

130  - 

\ 

A  T  &  S  F  pf 

101 

100* 

100*- 

* 

M  K  &  Tex 

39* 

38* 

39  - 

1 

At  C  Line 

129 

128* 

128*+ 

1 

M  K  &  T  pf 

701 

70* 

70f+ 

* 

Bait  &  Ohio 

118f 

118* 

118f- 

1 

Mo  Pacific 

88* 

86* 

88  + 

* 

Bklyn  R  T 

801 

78* 

79  - 

U 

Nat  Bis  Co 

85* 

84* 

85  + 

* 

Can  Pac 

190* 

188* 

190  + 

* 

Nat  Lead 

72 

7H 

72  + 

* 

Ches  &  Ohio 

52* 

511 

52f+ 

* 

N  Y  Air  Br 

135| 

135 

135!  + 

* 

Chi  &  N  W 

189* 

188 

189  + 

* 

N  Y  Central 

130* 

129* 

1291- 

| 

C  &  N  W  pf 

230 

230 

230 

Nor  Pacific 

160* 

158 

158*- 

* 

Chi  Gt  W 

161 

16 

16*- 

t 

Ont  &  Wes 

46* 

45* 

45*  + 

\ 

C  Gt  W  pf  B 

24* 

24* 

24* 

Ont  Silver 

7 

6| 

7 

C  M  &  St  P 

155* 

153f 

154f+ 

* 

Pac  Coast 

121* 

121 

121   - 

\ 

CMSP  sb  1  pd 

33| 

32* 

33 

Pacific  Mail 

38 

37* 

37*  + 

\ 

Chi  U  Tr  pf 

17* 

17* 

17*+ 

i 

Penn  RR 

1351 

135 

186*  + 

\ 

Col  Fuel  &  I 

64* 

53* 

54*+ 

i 

Reading  1  pf 

90 

90 

90  + 

\ 

Cons  Gas 

137* 

137 

137   - 

1 

Rock  Island 

271 

27 

27*+ 

\ 

Del  &  Hud 

216 

215* 

215*- 

* 

Rock  Isl  pf. 

62 

61 

61]  + 

If 

Del  L  &  W 

480 

480 

480  -32* 

St  P  &  Om  pf 

165 

165 

165  - 

](} 

Den  &  R  Gr 

39* 

37* 

38  - 

1 

Sou  Pacific 

96* 

94* 

95  + 

\ 

Den  &  R  G  pf 

81* 

80* 

80*- 

If 

Sou  Pac  pf 

117* 

117f 

117|- 

\ 

Erie 

40* 

88* 

39  - 

1 

Twin  CRT 

106* 

104 

105  - 

\\ 

Erie  1st  pf 

73f 

71f 

72*- 

H 

Union  Pac 

180 

179 

179*- 

* 

Erie  2d  pf 

63* 

62* 

88**- 

f 

U  S  Steel 

49f 

49* 

495  + 

f 

Gen  Elec  Co 

159 

157* 

157*  + 

i* 

U  S  Steel  pf 

107* 

106| 

107   + 

* 

304  PERCENTAGE  AND  ITS  APPLICATIONS         [§§783-784 

783.  To  find  the  market  value,  cost,  or  selling  price  of  stocks. 

784.  Examples.    1.    According  to  the  closing  quotation  given  in 
the  list  find  the  market  value  of  350  shares  American  Sugar  Refining 

Co.  stock. 

SOLUTION 

$  132.75  =  market  value  of  1  share, 

$  132.75  x  350  =  $46,462.50  =  market  value  of  350  shares. 

NOTE.   Unless  otherwise  specified,  the  par  value  of  one  share  is  $  100. 

#.   Find  the  cost  of  500  shares  of  Atlantic  Coast  Line  at  the 
closing  quotation;   brokerage  \%. 

SOLUTION 

$  100  =  the  par  value  of  1  share. 
$  128.50  =  the  market  or  quoted  value  of  1  share. 
\  %  of  $100  =  $.12£,  brokerage  on  1  share. 

$  128.62|  =  cost  of  1  share.     $  128. 62£  x  500  =  $  64,312.50  =  the  cost  of  500 
shares. 

8.   A  broker  sells  for  me  350  shares  Chicago   &  Northwestern 
at  the  highest  quotation;  usual  brokerage.      How  much  should  I 

receive  ? 

SOLUTION 

$189.50  x  350  =  $66,325,  the  amount  received  by  the  broker. 

$  100  x  350  =  $35,000,  the  par  value. 

£%  of  $35,000  =  $43.75,  the  brokerage. 

$66,325  -  $43.75  =  $66,281.25,  the  proceeds  of  the  sale. 

WRITTEN  EXERCISE 

Find  the  market  value  of  the  following  stocks  at  the  closing 
quotations  in  the  list,  p.  303 : 

1.  50  shares  American  Locomotive. 

2.  75  shares  American  Tobacco  preferred. 
8.   150  shares  Amalgamated  Copper. 

4-   56  shares  General  Electric  Co. 
5.   300  shares  Missouri  Pacific. 

6.   A  broker  sold  for  me  at  par  260  shares  Atchison,  Topeka, 
&  Santa  Fe ;  brokerage  -J-  %.     How  much  should  I  receive  ? 


§§  784-786]  STOCKS  305 

7.  Find  the  proceeds  of  650  shares  Manhattan  Railroad  stock 
at  the  closing  quotation  in  the  list,  p.  303 ;  brokerage  £  % . 

8.  How  much  must  be  sent  to  a  broker  in  order  that  he  may 
buy  60  shares  Iowa  Central  preferred  at  48^ ;  usual  brokerage  ? 

9.  What  is  the  total  par  value  of  300  shares  Canadian  Pacific ; 
150  shares  Denver  &  Rio  Grande  common ;  50  shares  Denver  &  Rio 

'  Grande  preferred ;  75  shares  National  Biscuit ;  80  shares  Ontario 
&  Western ;  80  shares  Rock  Island  common ;  and  70  shares  Rock 
Island  preferred?  What  is  the  total  market  value,  taking  each 
stock  at  the  closing  quotation  for  that  stock  in  the  list,  p.  303  ? 

10.  The  directors  of  a  rapid  transit  company  decide  to  increase 
their  stock,  by  declaring  a  stock  dividend  of  5  %  (781) ;  that  is, 
each  stockholder  is  presented  with  newly  issued  stock  to  the  extent 
of  5  %  of  his  holdings.  Just  before  this,  the  stock  sold  at  115 J. 
What  ought  to  be  the  market  value  afterwards? 

785.  To  find  the  number  of  shares  or  the  par  value  of  stock. 

786.  Example.     Sold  Consolidated  Gas  stock  at  137J  for  $  44,000, 
Find  the  number  of  shares  and  the  par  value  of  the  stock. 

SOLUTION 

$  137.50  =  the  market  value  of  1  share  of  stock. 
$  44,000  =  the  market  value  of  the  whole  number  of  shares, 
$  44,000  -f-  $  137.50  =  320,  the  number  of  shares  sold. 
$  100  x  320  =  $32,000,  the  par  value  of  the  stock. 

WRITTEN  EXERCISE 

1.  How  many  shares  of  Chicago  &  Northwestern  common  can 
be   bought   for   $75,850,  at   the   highest   quotation  on   page  303; 
brokerage  \  %  ?     Upon  what  sum  will  dividends  be  declared  ? 

2.  How  many  shares  of  Brooklyn  Rapid  Transit  must  be  sold 
at  the  lowest  quotation,  regular  brokerage,  to  bring  $39,187.50? 

S.  How  many  shares  Twin  City  Rapid  Transit  bought  at  the 
lowest  quotation  and  sold  at  the  highest  quotation  will  produce  a 
gain  of  $  1850 ;  usual  brokerage  both  ways  ? 

4.   A  broker  bought  on  his  own  account  Southern  Pacific  at  the 
highest  quotation,  for  $28,725.     How   many  shares  did  he  get? 
What  is  the  amount  of  a  4  %  dividend  on  them  ? 
MOORE'S  COM.  AR. — 20 


306  PERCENTAGE   AND   ITS   APPLICATIONS         [§§787-790 

787.  To  find  the  amount  of  investment. 

788.  Example.   What  sum  must  be  invested,  regular  brokerage, 
in  Twin  City  Rapid  Transit  preferred,  at  the  lowest  quotation  in 
the  list,  p.  303,  to  earn  $  1500  per  annum  if  the  annual  dividend  is 

4%?    ' 

SOLUTION 

The  dividend  on  1  share  =  $  4. 

$1500  -s- $4  =  375. 

Hence,  the  dividend  on  375  shares  =  $  1500. 

The  cost  of  1  share  =  $  104. 12£  ($  104  +  $  .12|  brokerage). 

$  104.12£  x  375  =  $39,046.88,  the  cost  of  375  shares. 

WRITTEN  EXERCISE 

1.  What  sum  must  be  invested  in  Delaware  &  Hudson,  at  the 
highest  quotation  in  the  list,  p.  303,  including  \  %  brokerage,  to  earn 
an  income  of  $2700  per  year,  the  average  annual  dividend  being 
9%? 

2.  If  Baltimore  &  Ohio  stock  pays  an  annual  dividend  of  6  %, 
how  much  must  be  invested  in  this  stock  at  118},  regular  brokerage, 
to  produce  an  annual  income  of  $  1500  ? 

3.  What  can  one  afford  to  pay,  exclusive  of  brokerage,  for  stock 
that  averages  an  annual  dividend  of  9  %  in  order  to  realize  5  %  on 
the  investment  ? 

4.  If  New  York  Central  pays  an  annual  dividend  of  6%,  and  you 
buy  through  a  broker,  at  J%,  enough  stock  to  give  you  an  income 
of  $  2400  per  year,  what  would  it  cost  you  at  the  quotation  129^  ? 

789.  To  find  the  rate  per  cent  of  income. 

790.  Example.    What  is  the  per  cent  of  income  on  an  investment 
in  Canadian  Pacific  at  the  closing  quotation  in  the  list,  p.  303,  regu- 
lar brokerage,  if  the  dividends  are  6  %  per  annum  ? 

SOLUTION 

$190.125  =  the  cost  of  1  share. 

$  6  =  the  income  on  1  share. 

$6  -=-  $  190.125  =  3.155  +  %,  the  rate  of  income. 


§§  790-71)2]  STOCKS  307 

WRITTEN   EXERCISE 

1.  If  American  Sugar  Refining  Co.  stock  yields  an  annual  divi- 
dend of  7  %,  what  is  the  rate  per  cent  of  income  on  the  investment, 
for  stock  bought  at  the  lowest  quotation  in  the  list,  page  303,  regular 
brokerage  ? 

2.  Which  is  the  better  investment  —  Reading  first  preferred,  at 
90,  paying  a  dividend  of  4  %  annually,  or  New  York  Air  Brake, 
at  135,  paying  8%  annual  dividends;  no  brokerage?     What  is  the 
difference  in  the  rate  of  income  ? 

3.  What  per  cent  of  income  will  you  receive  on  an  investment 
in  New  York  Central,  paying  an  annual  dividend  of  6  %,  if  pur- 
chased at  130^  through  a  broker  at 


791.  To  find  the  dividend  on  stocks. 

792.  Example.     The  Atchison,  Topeka  &  Santa  Fe  Railroad  Co. 
reported  a  net  income  of  $  29,701,795  for  one  year.     Their  common 
stock  was  $  102,707,000.     The  directors  declared  a  5  %  dividend  on 
this,  and  the  remainder  of  the  net  income  was  carried  to  surplus  fund. 
How  much  was  the  dividend?     How  much  was  the  surplus  fund 
amount  ?     If  you  owned  25  shares  of  this  stock,  what  would  be  your 

part  of  the  dividend  ? 

SOLUTION 

5%  of  $102,707,000  =  $5,135,350,  the  amount  of  the  dividend. 

$  29,701,795  -  $5,135,350  =  $24,566,445,  the  surplus  fund  amount. 

Dividends  are  declared  on  the  par  value  of  the  stock. 

The  par  value  of  your  stock  is  $2500. 

5%  of  $  2500  =  $  126,  the  dividend  due  you. 

WRITTEN   EXERCISE 

1.  You   paid   through   a  broker,  at  1%,  $30,870  for   Atlantic 
Coast  Line  stock  bought  at  the  closing  quotation  in  the  list,  page 
303.     What  annual  income  is  derived  if  3  %  semiannual  dividends 
are  declared  ? 

2.  The  common  stock  of  a  railroad  company  is  $  54,000,000,  and 
the  preferred  stock  (767)  is  $6,000,000.     The  directors  declare   a 
41-%  dividend  on  the  preferred  stock  and  a  3%  dividend  on  the 
common  stock.     What  is  the  surplus  if  the  total  net  earnings  are 
$  1,965,475.50  ? 


308  PERCENTAGE   AND   ITS  APPLICATIONS         [§§792-794 

S.  The  Southern  Pacific  has  a  common  stock  capitalization  of 
$  197,849,200,  and  declares  an  annual  dividend  of  5  % .  What  is  the 
total  dividend  and  how  much  is  due  C,  who  owns  22  shares  ? 

4.  The  Twin  City  Rapid  Transit  Co.,  capital  $20,100,000,  de- 
clared a  quarterly  dividend  of  1$%.     What  is  the  total  dividend, 
and  how  much  of  it  is  due  F,  who  owns  75  shares  ? 

5.  The  New  York  Air  Brake  Co.  is  capitalized  at  $10,000,000. 
If  its  net  earnings  for  a  year  are  $  1,345,308.25,  and  if  3J  %  of  the 
net  earnings  is  set  aside  as  a  surplus  fund,  an  8  %  dividend  is 
declared,   and   the   balance   is   carried  to   undivided   profits,  what 
amount    goes    to    surplus   fund,    dividend,   and   undivided   profits 
respectively  ? 

793.  To  find  the  rate  per  cent  of  dividend,  the  capital  stock  and  net 
earnings  being  given. 

794.  Example.     The  Canada  Southern  Railway  Co.  has  a  capi- 
talization of   $15,000,000.     If  its   net   earnings   are    $383,762.50, 
how  great  an  even  per  cent  dividend  may  be  declared  if  2  %  of  the 
net  earnings  are  to  go  to  surplus  fund  before  any  allowance  is  made 
for  dividend  ? 

SOLUTION 

2%  of  $.383,762.50  =  $7675.25  surplus. 

$383,762.50 -$7,675.25  =  $376,087.25,  the  amount  to  go  to  dividend  and 
undivided  profits. 

1  %  of  $  15,000,000  =  $  150,000. 

$376,087.25  +  $  150,000  =  2,  with  a  remainder  of  $76,087.25. 

Hence  a 2%  dividend  may  be  declared,  and  $76,087.25  will  be  left  as  undi- 
vided profits. 

WRITTEN  EXERCISE 

1.  The  Delaware  &  Hudson  Co.  has  a  capitalization  of  $42,250,- 
500.     If  its  net  earnings  are  $  3,802,545,  how  large  an  even  rate  per 
cent  dividend  may  be  declared  ? 

2.  The  common  stock  of  the  Baltimore  &  Ohio  Railroad  Co.  is 
$  152,165,500.     If  its  net  earnings  for  a  year  are  $  9,141,065.60,  what 
is  the  greatest  even  per  cent  of  dividend  that  may  be  declared  and 
what  balance  will  there  be  for  undivided  profits  ? 


§§  794-796] 


STOCKS 


309 


3.  Suppose  the  Canadian  Pacific  Railway  Co.  has  gross  earnings 
of  $  22,973,312,  and  expenses  of  $  8,350,545.  If  its  capitalization  is 
$105,307,100,  what  even  per  cent  dividend  may  be  declared  and 
what  will  be  the  amount  of  undivided  profits  if  2J%  of  the  net 
earnings  is  first  set  aside  as  a  surplus  fund  ? 

795.  To  find  the  profit  or  loss  when  a  person  buys  and  sells  stocks 
on  a  margin. 

796.  Example.    On  Jan.    5,  a  speculator   deposited  with  his 
broker   $  1500  as  a  margin.     The  broker  purchased  for  him  150 
shares  Southern  Pacific  at  90.     On  Jan.  17  the  broker  sold  the  stock 
at  the  highest  quotation  in  the  list,  p.  303.    What  was  the  cus- 
tomer's profit;  interest  6%,  brokerage 


SOLUTION 


(2^^^ 


.faa 


$  2300.96  -  $  1500  =  $800.96,  the  amount  gained. 


WRITTEN  EXERCISE 

1.  On  Jan.  11  a  speculator  deposited  with  his  broker  $  3000  as  a 
margin,  and  through  him  bought  150  shares  Chesapeake  &  Ohio  at 
48,  and  150  shares  Missouri  Pacific  at  83.  On  Jan.  17  the  broker 
sold  the  stock  at  the  highest  quotation  in  the  list,  p.  303.  How 
much  did  the  speculator  gain  ;  interest  at  6  %,  brokerage 


g.  On  Jan.  7  W  deposited  $  2000  as  margin  with  his  broker,  who 
bought  for  him  100  shares  Delaware  &  Hudson  at  217,  and  100 
shares  Baltimore  &  Ohio  at  120.  On  Jan.  17  the  stock  was  sold  at 
the  closing  quotations  in  the  list,  p.  303.  What  was  W's  gain  or 
loss  ;  interest  6  °/0  >  brokerage  ^  %  ? 


310  PERCENTAGE   ANDT  ITS  APPLICATIONS         [§§796-800 

8.  On  Jan.  17  a  speculator  deposited  $4000  as  a  margin,  and  by 
his  orders  the  broker  bought  200  shares  Atchison,  Topeka  &  Santa 
Fe  at  108J,  100  shares  Atlantic  Coast  Line  at  133|,  and  100  shares 
Chicago  &  Alton  at  27J.  On  Jan.  25  (the  margin  being  nearly  ex- 
hausted) the  broker  "  sold  him  out  "  at  the  following  quotations  : 
Atchison,  Topeka  &  Santa  Fe  at  98f,  Atlantic  Coast  Line  at  119|, 
and  Chicago  &  Alton  at  24J.  How  much  did  the  speculator  lose  ; 
interest  6  %,  brokerage 


BONDS 

797.  A  bond  is  a  written  or  printed  obligation  under  seal  issued 
by  a  company  or  corporation,  municipal  or  state  government,  or  by 
the  federal  government.     It  is  conditioned  to  pay  a  certain  sum  of 
money  at  a  specified  time  and  at  a  fixed  rate  of  interest,  payable  at 
regular  intervals. 

Bonds  of  business  corporations  are  usually  secured  by  mortgages  on  their 
real  estate.  Municipal  bonds  are  issued  by  vote  of  the  people  or  their  repre- 
sentatives, and  for  their  payment  a  sinking  fund  is  accumulated  by  a  yearly  rate 
per  cent  levied  on  all  the  real  property  within  the  limits  of  the  municipality. 

798.  Government  bonds  are  bonds  issued  by  the  federal  govern- 
ment.    Their  names  are  usually  derived  from  the  interest  they  bear 
and  the  time  when  due. 

Thus  "  U.S.  4's,  1912,"  is  understood  to  mean  "United  States  bonds  bearing 
4%  interest,  and  due  in  1912";  and  "U.S.  3's,  1925,"  is  understood  to  mean 
"  United  States  3%  bonds  due  in  1925." 

799.  A  coupon  bond  is  a  bond  that  has  coupons  or  certificates  of 
interest  attached.     When  the  interest  becomes  due,  these  coupons 
are  detached  and  surrendered  upon  receipt  of  the  interest  repre- 
sented by  them. 

The  interest  coupons  on  government  coupon  bonds  are  payable  to  the 
bearer,  and  will  be  cashed  by  any  bank  or  banker  in  the  United  States.  Coupon 
bonds  may  be  converted  into  registered  bonds  of  the  same  issue. 

800.  A  registered  bond  is  one  which  is  payable  to  the  owner  as 
registered  in  the  books  of  the  corporation  or  government  issuing  it. 
Registered  bonds  can  be  transfered  only  by  assignment  and  registry 
on  the  books. 

The  interest  on  registered  bonds  is  paid  by  checks,  payable  to  the  order  of 
the  registered  owner,  and  sent  to  him.  The  checks  for  interest  on  government 
bonds  are  readily  cashed  by  banks  and  bankers. 


§§  801-8C3] 


BONDS 


311 


801.  As  with  stocks,  the  par  value  of  bonds  is  their  face  value ; 
the  market  value  is  the  amount  at   which   they  are  quoted  in  the 
market.     Bonds  are  above  par  or  at  a  premium  when  they  are  worth 
more  than  their  face  value ;  below  par  or  at  a  discount  when  they  are 
worth  less  than  their  face  value. 

802.  Bond  quotations  are  the  market  prices  or  rates  that  the 
bonds  sell  for. 

The  income  from  bonds,  unlike  that  from  stocks,  is  fixed;  that  is,  it  is 
in  no  way  affected  by  the  general  conditions  of  the  corporation,  so  long  as  the 
corporation  is  solvent. 

Bonds  are  usually  quoted  flat;  that  is,  the  quoted  price  is  for  the  bond  as  it 
is  at  the  time  of  the  quotation,  including  accrued  interest,  except  that  after  the 
closing  of  the  books  registered  bonds  are  quoted  less  the  interest.  The  interest 
then  due  belongs  to  the  holder  of  the  bonds  at  the  time  the  books  are  closed. 

803.  The  list  herewith  shows  part  of  one  day's  bond  sales  (so 
many  dollars  par  value  of   each)  on  the  floor  of  the  New  York 
Stock  Exchange ;  it  is  taken  from  the  New  York  Sun,  Wall  Street 
edition,  under  date  of  Jan.  17,  1907 : 


Adams  Exp  4s 
500            102^ 

Chi    &    E   111   s  f 

6's 

111  Central  4s  1953 
2000            103| 

Penna  cv  3£s 
20000             96| 

3000  102* 

1000            lOOf 

111  Cent  L  div  3|s 

1000              96| 

Am  Ice  deb  6s 

Chi  Mil  &  St  P  4s 

1000              89f 

Penna  3£s  1915 

1000     ....   89 

1000            106 

3000  87  J 

20000              92£ 

Am  Tobacco  6s 

Chi  R  I  &  Pac  RR 

Mo  K  &  T  s  f  4|s 

22000  93 

3000  110J 

gold  5s 

19000          .   87£ 

3000              92| 

39000  110 

1000  90£ 

Mo  Pacific  5s  1920 

24000  93 

2000  110£ 

3000          .  90f 

1000     .  .  .   105 

20000              92| 

Am  Tobacco  4s 

2000  90£ 

N  Y  Central  3^s 

2000  93 

3000  77f 

5000              90 

4000              93  \ 

7000              93| 

6000          .  78 

Erie  c  v  4s  ser  A 

58000              93  \ 

Union  Pacific  4s 

registered 

10000            100  £ 

North  Pacific  3s 

2000            101^ 

500  75| 

5000  1004 

4000  73| 

4000s  15.  .  .10  If 

Bait  &  Oh  gold  4s 
16000  101£ 

15000  100 
37000  99£ 

5000  73£ 
5000  73£ 

U  S  3s  cpn 
3500  103 

Ches  &  Ohio  6s 

2000       .  .  .  99| 

2000              73 

U  S  Steel  s  f  5s 

1000  115 

10000  100 

Or  S  L  fdg  4s 

6000  98£ 

Ches  &  Ohio  4|s 

69000  ....  991 

4000              94 

17000              98  \ 

1000  104£ 

Erie  prior  lien  4s 

-    5000  94} 

4000  98£ 

Chi  &  Alton  3£s 

4000  971, 

Penna  4£s  1921 

11000              98  l 

4000              76 

Green  Bay  &  West 

1000            107 

11000              98£ 

Chi  Bur  &  Q  4s 

deb  ser  B 

3000  106| 

1000  981 

2000..       .   96i 

2000..     .  m 

3000..      ..1061 

19000..        .   984 

312  PERCENTAGE   AND   ITS  APPLICATIONS  [§804 

NOTE.  Brokers  usually  charge  ^%  brokerage  for  transactions  in  bonds. 
That  rate  is  to  be  understood  if  none  is  specified. 

804.  Example.  What  rate  per  cent  per  annum  interest  will 
Chicago  &  Alton  3j's  yield  on  the  investment,  if  bought  through  a 
broker  at  the  price  quoted  in  the  list,  p.  311  ? 

SOLUTION 

$76  -f  .125  =  cost  of  $  100  par  value  of  the  bonds. 

$3.50  =  the  income  on  $100  worth  of  the  bonds. 

$8.60  -*-  $76.125  (cost  of  bonds)  =  4.5977+  %,  the  rate  of  income. 


WRITTEN  EXERCISE 

1.  If  a  broker  invested  on  his  own  account  in  Chesapeake  6c 
Ohio  5's  as  quoted  in  the  list,  p.  311,  what  per  cent  of  income  would 
he  receive  ? 

2.  Find  the  proceeds  of  the  United  States  3's  coupon  sold  through 
a  broker. 

S.   How  much  must  be  invested  in  Chesapeake  &  Ohio  4±-'s  to 
produce  a  semiannual  income  of  $1350;  regular  brokerage? 

4.  What  per  cent  income  will  be  produced  by  $  358,750  invested 
in  Adams  Express  Company's  4's  at  the  market  quotation,  allowing 
the  regular  brokerage? 

5.  How  much  must  you  invest  through  a  broker  in  Pennsyl- 
vania 4i-'s,  1921,  at  the  last  quotation,  so  that  you  may  have  an 
income  of  $2700  per  year?     This  income  is  what  per  cent  of  the 
investment  ? 

6.  A  sells  through  a  broker  one  $  5000  Pennsylvania  3£,  1915, 
at  the  last  quotation,  and  loans  the  proceeds  at  5%.     How  much 
will  his  yearly  income  thereby  increase? 

7.  How  many  $  1000  Illinois  Central  4's,  1953,  bought  at  98J  and 
sold  at  the  list  quotation,  will  yield  $2375  gain,  usual  brokerage 
both  ways  ? 

8.  A  has  an  annual  income  of  $  880  on  an  investment  in  Balti- 
more &  Ohio  $500  gold  4's.     How  many  does  he  own?     If  they 
were  bought  at  the  quotation  in  the  list,  through  a  broker,  what 
rate  per  cent  per  annum  does  he  receive  on  his  investment  ? 


§  804]  BONDS  318 

9.  Which  would  be  the  better  investment,  Erie  convertible  4's, 
series  A,  at  the  last  quotation  in  the  list,  or  Missouri  Pacific  5's, 
1920,  if  both  were  purchased  through  a  broker?  How  much  better? 


WRITTEN  REVIEW 

1.  What  will  be  the  cost,  including  \  °/0  brokerage,  of  250  shares 
Denver  &  Rio  Grande,  300  shares  Atchison,  Topeka  &  Santa  Fe, 
50  shares  New  York  Central,  40  shares  Ontario  &  Western,  125 
shares  Louisville  &  Nashville,  150  shares  American  Tobacco  pre- 
ferred, all  at  the  closing  quotations  on  the  list,  p.  303? 

2.  What  annual  income  is  derived  from  investing  $27,500,  exclu- 
sive of  brokerage,  in  Consolidated  Gas  at  137^,  if  it  averages  5  % 
annual  dividends  ? 

3.  How  much  must  be  invested,  exclusive  of  brokerage,  in  Amal- 
gamated Copper  at  115J,  so  that  an  annual  income  of  $  2500  may  be 
realized  if  a  4  %  yearly  dividend  is  declared  ? 

4.  A  bought  through  his  broker,  at  |-%,  500  shares  Pacific  Coast, 
for  which  he  paid  the  broker  $  60,812.50.     With  what  market  quo- 
tation does  the  price  he  paid  agree  ?     How  much  was  the  brokerage? 

5.  In  example  1  if  the  average  annual  dividend  was  41  %,  what 
was  the  rate  per  cent  of  interest  on  the  investment? 

6.  In  example  2  what  is  the  rate  per  cent  interest  on  the  invest- 
ment ? 

7.  On  Jan.  2  I  deposited  with  my  broker  $6000  as  a  margin, 
and  he  bought  for  me  250  shares  Erie  at  40 f,  200  shares  Missouri 
Pacific  at  88^,  and  150  shares  Brooklyn  Rapid  Transit  at  82.     On 
Jan.  17  the  stocks  were  quoted  late  in  the  day  at  the  closing  figures 
in  the  list,  p.  303.     How  much  must  I  deposit  to  make  my  margin 
good ?     If  the  broker  had  "  sold,  me  out"  because  I  could  not  make 
my  margin  good,  how  much  would  I  have  lost  ?  ' 

8.  A   sold  500   shares  Louisville  &  Nashville   stock  at  135J, 
through  a  broker,  and  bought  with  the  proceeds  of  the  sale  Chicago, 
Milwaukee  &  St.  Paul  4  %  bonds  at  106,  through  a  broker.     How 
many  $500  bonds  did  he  get,  and  how  much  unexpended  balance 
was  there  due  him  ? 


314  PERCENTAGE  AND   ITS   APPLICATIONS  [§§804-806 

9.  Sold  three  $  1000  American  Tobacco  6's  at  110J,  and  with 
the  proceeds  bought  Northern  Pacitic  at  158.  Later  in  the  day 
I  sold  the  stock  at  160J.  How  much  did  I  gain,  allowing  the 
usual  brokerage  on  all  the  transactions  ?  How  much  did  the  broker 
have  belonging  to  me  ? 

10.  American  Ice  Securities  pays  an  annual  dividend  of  7  %  ; 
Delaware  &  Hudson,  9%;    Baltimore  &  Ohio,  6%;  Erie,  2d  pre- 
ferred, 4%.     If  bought  at  the  closing  quotations  in  the  list,  p.  303, 
with  no  brokerage,  what  is  the  rate  per  cent  of  income  on  each  ? 

11.  The  American  Smelting  Kefining  Co.  had  reported  net  earn- 
ings; during  the  fiscal  year  1906,  of  $  10,161,358.     Its  common  stock 
was  $50,000,000.     If  5%  of  the  net  earnings  is  set  aside  as  surplus 
fund,  a  7  %  dividend  is  declared,  and  the  balance  carried  to  undi- 
vided profits,  what  sums  go  respectively  to  surplus  fund,  to  divi- 
dend, to  undivided  profits  ? 

12.  What  sum  invested  in  Chicago  &  Eastern  Illinois  sinking  fund 
6's  at  100 1  will  produce  a  yearly  income  of  $3000,  no  brokerage  ? 

18.  X  owns  200  shares  Reading,  1st  preferred,  which  cost  him 
$  18,000.  He  realizes  annually  5  %  on  his  investment.  What  rate 
of  dividend  was  declared  ? 

H.  The  net  earnings  of  the  Canadian  Pacific  for  the  fiscal  year 
1906  were  $22,973,312,  and  the  capital  stock  was  $105,307,100.  If 
50  °/o  of  the  net  earnings  is  carried  to  surplus  fund,  what  even  per 
cent  of  dividend  may  be  declared,  and  how  much  will  be  left  as 
undivided  profits  ? 

15.  Y  owned  300  shares  of  the  Canadian  Pacific  stock.  Z  owned 
250  shares.  If  Y  bought  his  stock  at  188f,  and  Z  bought  his  stock 
at  190,  what  is  the  ra,te  per  cent  of  income  on  each  man's  invest- 
ment, making  no  allowance  for  brokerage  ?  What  does  each  receive 
as  dividend  ? 

INSURANCE 

805.  Insurance  treats  of   those  computations  arising  from  con- 
tracts guaranteeing  security  against  loss  or  damage. 

806.  The  parties  to  insurance  are  the  insured  or  assured  and  the 
insurer  or  underwriter. 


§§  807-817]  INSURANCE  815 

807.  The  insured  or  assured  is  the  person  protected,  or  insured, 
against  loss  or  damage. 

808.  The  insurer  or  underwriter  is  the  party  that   guarantees 
security  against  loss  or  damage. 

Insurers  or  underwriters  are  usually  incorporated  companies. 

809.  A  policy  is  a  written  contract  between  the  insured  and  the 
insurer.     It  sets  forth  the  conditions  under  which  the  risk  is  taken, 
the  liability  of  the  insurance  company,  the  time  the  insurance  is  to 
continue,  the  premium. 

810.  A  valued  or  closed  policy  is  one  in  which  a  fixed  value  is 
given  to  the  thing  insured. 

A  valued  or  closed  policy  is  the  ordinary  form  used  in  general  fire  insurance. 

811.  An  open  policy  is  one  in  which  no  fixed  value  is  given  to 
the  thing  insured.     In  an  open  policy  additional  insurance  may  be 
entered  at  any  time  at  rates  and  under  conditions  agreed  upon. 

812.  The  premium  is  the  sum  paid  for  insurance. 

813.  The  term  of  insurance  is  the  period  of  time  for  which  the 
risk  is  taken  or  the  property  insured. 

814.  Premium  rates  are  sometimes  given  as  a  specified  number 
of  cents  per  $  100,  and  sometimes  as  a  certain  per  cent  of  the  sum 
insured.     They  depend  upon  the  nature  of  the  risk  and  the  length 
of  time  for  which  the  policy  is  issued. 

Insurance  is  usually  effected  for  a  year  or  a  term  of  years. 

815.  Short  rates  are  certain  rates  of  premium  charged  by  insur- 
ance companies  for  terms  less  than  one  year.     Short  rates  are  pro- 
portionately higher  than  yearly  rates. 

816.  An  insurance  agent  is  one  who  acts  for  an  insurance  com- 
pany in  obtaining  insurance,  collecting  premiums,  adjusting  losses, 
reinsuring,  etc. 

817.  An  insurance  broker  is  a  person  who  negotiates  insurance 
for  others,  for  which  he  receives  a  brokerage  from  the  company 
taking  the  risk ;  he  is  considered,  however,  an  agent  of  the  insured, 
not  of  the  company. 


316  PERCENTAGE   AND   ITS  APPLICATIONS         [§§  818-825 

818.  Insurance  companies  are  distinguished  by  the  way  in  which 
they  are  organized;  as  stock  companies,  mutual  companies,  and  mixed 
companies. 

819.  A  stock  insurance  company  is  one  whose  capital  has  been 
contributed  and  is  owned  by  the  stockholders,  who  share  the  gains 
and  are  liable  for  the  losses. 

820.  A  mutual  insurance  company  is  one  in  which  the  gains  and 
losses  are  shared  by  the  insured  parties. 

821.  A  mixed  insurance  company  is   one  which  combines  the 
features  of  both  stock  and  mutual  companies. 

In  mixed  companies  all  gains  above  a  limited  dividend  to  the  stockholders 
are  divided  among  the  policy  holders. 


PROPERTY  INSURANCE 

822.  Property  insurance  is  the  insurance  of  property  against  any 
specified  casualty. 

823.  Property  insurance  includes : 

1.  Fire  insurance,  or  indemnity  for  loss  of,  or  damage  to,  property 
by  fire. 

2.  Marine  insurance,  or  indemnity  for  loss  of,  or  damage  to,  a 
ship  or  its  cargo  by  any  specified  casualty  at  sea  or  on  inland  waters. 

3.  Live  stock  insurance,  or  indemnity  for  loss  of,  or  damage  to, 
horses,  cattle,  etc.,  and  from  lightning  or  other  casualty. 

4-  Transit  insurance,  or  indemnity  for  loss  of,  or  damage  to,  goods 
transported  from  one  place  to  another  by  land  or  by  both  land  and 
water. 

824.  Insurance  policies  are  sometimes  classified  as  ordinary  poli- 
cies and  average  clause  policies. 

825.  Under  an  ordinary  policy  the  company  will  pay  the  full 
amount  of  any  loss  or  damage  that  does  not  exceed  the  sum  covered 
by  the  policy. 

Thus,  if  a  house  worth  $  12,000  is  insured  for  $  9000,  and  a  fire  occurs  by 
which  a  loss  of  $  7000  is  sustained,  the  company  is  bound  to  pay  the  full  loss,  or 


§§  825-832]  INSURANCE  317 

$  7000 ;  but  if  the  loss  should  be  $  10,000,  or  any  sum  in  excess  of  $  9000,  the 
company  will  pay  only  the  $  9000  specified  in  the  policy. 

826.  Under  an  average  clause  policy  the  company  will  pay  only 
such  a  proportion  of  the  loss  as  the  policy  is  of  the  entire  value  of 
the  thing  insured. 

Thus,  if  a  vessel  valued  at  $  12,000  is  insured  for  $  8000,  and  a  fire  occurs  by 
which  a  loss  of  $6000  is  sustained,  the  company  will  pay  two  thirds  (i^°o°(j)  of 
$  0000,  or  $  4000  ;  but  if  the  loss  is  total,  the  company  will  pay  the  full  $  8000, 
which  is  two  thirds  of  the  entire  valuation,  $  12,000. 

827.  Marine   insurance    policies    usually   contain    the   average 
clause. 

828.  Almost  all  insurance  companies  will  not  issue  a  policy  above 
a  certain  fixed  sum  ;  and  they  will  issue  only  one  policy  covering  the 
same  property.     Therefore,  if  a  person  owns  a  valuable  building,  he 
must  ordinarily  have  it  insured  in  several  different  companies,  in 
order  to  protect  his  interests. 

829.  If  property  that  is  insured  in  several  companies  is  damaged 
by  fire  to  the  extent  of  the  total  amount  of  the  insurance,  each  com- 
pany must  pay  the  full  amount  of  its  policy.     If  the  loss  is  less  than 
the  total  amount  of  the  insurance,  each  company  must  pay  such  a 
portion  of  the  loss  as  its  policy  is  a  part  of  the  entire  insurance. 

830.  To  cancel  a  policy  is  to  annul   the   contract  between   the 
insurer  and  insured. 

In  case  a  policy  is  terminated  at  the  request  of  the  insured,  he  is  charged  the 
short  rate  premium.  If,  however,  it  be  terminated  at  the  option  of  the  company, 
the  lower  long  rate  will  be  charged,  and  the  company  will  refund  the  premium 
for  the  unexpired  time  of  the  policy. 

831.  Salvage  is  an  allowance  made  to  those  rendering  voluntary 
aid  in  saving  vessels  or  cargoes  from  marine  casualties. 

Insurance  companies  usually  reserve  the  privilege  of  rebuilding,  replacing, 
or  repairing  damaged  property. 

832.  Computations  in  property  insurance  are  made  in  accord- 
ance with  the  general  principles  of  abstract  percentage,  the  amount 
insured  corresponding  to  the  base  ;  the  rate  of  the  premium  to  the 
rate;  and  the  premium  to  the  percentage. 


318  PERCENTAGE   AND   ITS  APPLICATIONS         [§§  832-836 

DRILL  EXERCISE 
1.   Find  the  cost  of  insuring  a  barn  and  contents  for  $4000  at 


2.  At  2%,  what  amount  of  insurance  can  I  procure  for  $  74? 

3.  If  $25  is  paid  for  insuring  property  worth  $1000,  what  is 
the  rate  ? 

4.  State  a  formula  for  finding  the  premium  when  the  amount 
insured  and  the  rate  of  premium  are  given. 

5.  Given  the   premium   and   rate   of   premium,  how   may   the 
amount  of  insurance  be  found? 

6.  Given  the  premium  and  the  amount  insured,  how  may  the 
rate  of  premium  be  found  ? 

7.  A  dealer  paid  $  125  for  insuring  a  cargo  of  grain  at  1\%  on 
|  of  its  value.     Find  the  value  of  the  grain. 

833.  To  find  the  cost  of  insurance. 

834.  Example.     How  much  will  it  cost  to  insure  a  store  and 
contents  for  $42,000  at  l\%  ? 

SOLUTION 

$42,000  =  the  amount  insured. 

$42,000  =  $630,  the  premium  charged. 


WRITTEN  EXERCISE 

1.  A  store  is  valued  at  $12,000  and  the  contents  at  $18,000. 
Find  the  cost  of  insuring  f  of  the  value  of  the  store  at  f  %,  and  f 
of  the  value  of  the  contents  at  f  %. 

2.  An  insurance  company,  having  insured  a  block  of  buildings 
for  $  200,000  at  75^  per  $  100,  reinsured  $  60,000  with  another  com- 
pany at  |%,  and  $80,000  with  another  at  f  %.     What  amount  of 
premium  did  it  receive  more  than  it  paid  ? 

835.  To  find  the  rate  of  insurance. 

836.  Example.     I  paid  $  30  for  insuring  a  house  worth  $  6400  at 
|  valuation.     What  was  the  rate  ? 

SOLUTION 

f  of  $6400  =  $4800,  the  face  of  the  policy. 

$  30  -r-  4800  =  .00625,  or  f  ,  the  rate  of  insurance. 


§§836-838]  INSUKANCE  319 

WRITTEN  EXERCISE 

1.  The  cost  of  insuring  f  of  a  cargo  of  wheat  worth  $  24,000  was 
$  240.     What  was  the  rate  of  insurance  ? 

2.  I  insured  f  of  a  stock  of  goods  worth  $  4500,  and  paid  $  18 
premium.     What  was  the  rate  of  insurance  ? 

837.  To  find  the  amount  insured. 

838.  Example.     A  man  paid  $280  to  insure  a  stock  of  goods  for 
3  months.     If  the  rate  of  premium  was  J%,  for  what  amount  was 
the  policy  issued? 

SOLUTION 

Let  100  %  represent  the  amount  of  the  policy. 

$280  =  the  premium  paid. 

|  %  =  the  rate  of  premium.    . 

Therefore|%=  $280. 

t%=*40;  |%orl%=$320. 

100%  =  $32,000,  the  amount  of  the  policy. 

WRITTEN  EXERCISE       ft 

1.  A  gentleman  paid  $  35.60  per  annum  for  insuring  his  house 
at  2%  on  -f  of  its  value.     What  was  the  value  of  the  house?    £f£/-J 

2.  A  ranchman  paid  a  premium  of  $  76.00  for  insuring  f  of  his 
herd  of  cattle  at  60^  per  $100.     If  the  cattle  were  valued  at  $40 
per  head,  how  many  had  he  ?      /^  - 

3.  The  contents  of  a  factory  were  insured  for  a  certain  sum  at 
IY%.     Later  the  goods  were  damaged  by  fire  and  losses  paid  by  the 
company  to  the  amount  of  $  18,750,  which  was  f  of  the  amount  in- 
sured.    If  the  amount  insured  was  |  of  the  value  inventoried,  what 
was  the  total  value  of  the  goods  ?      /,*    ^^ 

WRITTEN  REVIEW 

1.  Find  the  cost  of  insuring  a  cargo  of  wheat  valued  at  $  24,000 
at  li%. 

2.  How  much  insurance,  at  1£%,  can  be  procured  for  $  90  ? 

8.  If  it  cost  $  663  to  insure  a  certain  block  for  $  44,200,  what 
will  be  the  cost,  at  the  same  rate,  to  insure  a  block  valued  at 
$  105,000  if  $  1.50  extra  be  charged  for  the  policy  in  the  latter  case  ? 


320  PERCENTAGE   AND   ITS  APPLICATIONS  [§838 

4.  How  much  will  it  cost  to  insure  a  factory  for  $  42,000  at  f  %, 
and  its  machinery  for  $16,500  at  1J%,  the  charge  for  policy  and 
survey  being  $  2.50  ? 

5.  The  premium  on  a  cargo  of  3000  tons  of  coal  valued  at  $  3.50 
per  ton,  and  insured  at  f  of  its  valuation,  is  $  47.25.     Find  the  rate 
of  insurance.  ^L^ 

6.  If  a  store  and  its  contents  are  valued  at  $  27,000,  for  how 
much  must  it  be  insured  at  1^-%?  to  cover  loss  and  premium  in  case 
of  total  destruction  ?        /\   u\\   \v\ 

7.  A  cargo  of  teas,  valued  at  $  33,000,  was  insured  for  $  18,000 
in  a  policy  containing  an  "average  clause."     In  case  of  damage  to 
the  amount  of  $  21,000,  how  much  should  the  company  pay  ? 

*/  8.  The  steamer  Norseman,  valued  at  $  90,000,  is  insured  for 
$75,000,  at  21%.  What  will  be  the  actual  loss  to  the  insurance 
company  in  case  the  steamer  is  damaged  to  the  amount  of  $  20,000  ? 
9.  A  speculator  bought  2000  barrels  of  flour,  and  had  it  insured 
for  80%  of  its  cost,  at  3J%,  paying  a  premium  of  $429.  At  what 


price  per  barrel  must  he  sell  the  flour,  to  make  a  net  profit  of 
i^X    10.   I  insured  my  grocery  store,  valued  at  $  13,500,  and  its  con- 
tents, valued  at  $  33,000,  and  paid  $  350  for  premium  and  policy. 
If  the  policy  cost  $  1.25,  what  was  the  rate  per  cent  of  premium  ? 

AQ  11.  A  canal  boat  load  of  8400  bushels  of  wheat,  worth  90^  per 
bushel,  is  insured  for  f  of  its  value,  at  If  %  premium.  In  case  of 
the  total  destruction  of  the  wheat,  how  much  will  the  owner  lose  ? 

12.  A  stock  of  goods  valued  at  $  30,000  was  insured  for  18 
months,  at  1£%  ;  at  the  end  of  12  months  the  owner  surrendered 
the  policy.  If  the  "short  rate"  for  6  months  was  65^  per  $100, 
what  should  be  the  return  premium  ? 

18.  The  German  Insurance  Company  insured  the  Field  block  for 
$  105,000,  at  60^  per  $  100;  but  thinking  the  risk  too  great,  it  rein- 
sured $40,000  in  the  Home,  at  f  %,  and  $45,000  more  in  the  Mutual, 
at  J%.  How  much  premium  did  each  company  receive  ?  What  was 
the  gain  or  loss  of  the  German  ?  What  per  cent  of  premium  did  it 
receive  for  the  part  of  the  risk  not  reinsured  ? 

14.  A  block  of  stores  and  contents  was  insured  for  $  220,000  and 
became  damaged  by  fire  and  water  to  the  amount  of  $150,000.  Of 
the  risk,  $  40,000  was  taken  by  the  Hartford  Co.,  $  65,000  by  the 


§§838-844]  INSURANCE  321 

Manhattan,  $  35,000  by  the  ^Etna,  and  the  remainder  was  divided 
equally  between  the  Phoenix  and  the  Provident.  What  was  the  net 
loss  of  each  company,  if  the  premium  paid  was  1J  %  ? 

15.  A  factory  worth  $45,000  is  insured,  with  its  contents,  for 
$  62,500 ;  $  30,000  of  the  insurance  is  on  the  building,  $  12,500  on 
machinery  worth  $20,000,  and  $20,000  on  stock  worth  $35,000. 
A  fire  occurs  by  which  the  building  and  the  machinery  are  both 
damaged,  each  to  the  amount  of  $  15,000,  and  the  stock  is  entirely 
destroyed.  How  much  is  the  claim  against  the  company,  if  the 
risk  is  covered  by  an  "  ordinary  "  policy  ?  How  much  if  the  policy 
contains  the  "  average  clause  "  ? 

PERSONAL  INSURANCE 

839.  Personal  insurance  is  the  insurance  of  person.     It  includes : 

1.  Life  insurance,  or  indemnity  for  loss  of  life. 

2.  Accident    insurance,   or   indemnity   for   loss   from  disability 
occasioned  by  accident. 

3.  Health  insurance,  or  indemnity  for  loss  occasioned  by  sickness. 

840.  Life   insurance  policies  are  usually  either   life  policies  or 
endowment  policies. 

841.  A  life  policy  agrees  to  pay  to  the  beneficiary  named  in  it 
a  fixed  sum  of  money  on  the  death  of  the  insured. 

The  beneficiary  is  the  one  to  whom  the  insurance  is  guaranteed  to  be  paid. 

842.  An  endowment  policy  guarantees  the  payment  of  a  fixed 
sum  of  money  at  a  specified  time,  or  at  death,  if  the  death  occurs 
before  the  specified  time. 

843.  Life  insurance  companies  are  known  as  stock,  mutual,  mixed, 
and  cooperative.     Losses  sustained  by  stock  and  mixed   companies 
are  paid  either  from  reserve  funds,  or  by  assessment  on  the  stock- 
holders ;  those  sustained  by  mutual  and  cooperative  companies  are 
paid  by  pro  rata  or  fixed  contributions  of  the  policy  holders. 

844.  Life  insurance  may  be  made  payable  to  any  one  named  by 
the  insured.     If  made  payable  to  himself,  at  his  death  it  becomes  a 
part  of  his  estate  and  is  liable  for  his  debts ;  if  payable  to  another, 

MOORE'S  COM.  AR.  —  21 


822 


PERCENTAGE   AND  ITS  APPLICATIONS         [§§844-847 


that  other  cannot  be  deprived  of  the  benefit  of  the  insurance,  either 
by  the  will  of  the  person  taking  out  the  insurance,  or  by  his  creditors. 

845.  Any  one  having  an  insurable  interest  in  the  life  of  another 
may  take  out,  hold,  and  be  benefited  by,  a  policy  of  insurance  upon 
the  life  of  that  person ;  and  any  one  may  take  out  a  policy  in  his 
own  name  and  then  assign  it  to  any  creditor  or  to  any  one  having 
an  insurable  interest. 

846.  The  following  table  shows  the  rates  of  one  of  the  leading 
life  insurance  companies : 

ANNUAL  PREMIUMS  FOR  AN  INSURANCE  OF  $1000 


LIFE  POLICIES,  PAYABLE  AT  DEATH  ONLY 

ENDOWMENT  POLICIES,  PAYABLE  AS  INDI- 
CATED OR  AT  DEATH,  IF  PRIOR 

& 

< 

Continuous 
Premiums 

10 

Premiums 

15 

Premiums 

20 

Premiums 

<a 
tt 

<5 

In 
10  Yrs. 

In 
15  Yrs 

In 
20Yrs. 

In 
25  Yrs. 

In 
30  Yrs. 

In 
35Yrs. 

21 

$18  40 

$46  30 

$34  19 

$28  25 

21 

$101  53 

$6543 

$47  75 

$3745 

$30  86 

$26  41 

22 

18  80 

47  00 

34  71 

28  69 

22 

101  60 

65  51 

47  84 

37  55 

30  97 

•2«  55 

23 

19  28 

47  73 

35  -26 

29  15 

23 

101  68 

65  6(1 

47  94 

37  66 

31  10 

26  71 

24 

19  67 

48  47 

85  82 

29  63 

24 

101  76 

65  69 

48  04 

87  78 

81  24 

26  8S 

25 

20  14 

49  24 

36  40 

30  12 

25 

101  85 

65  79 

48  15 

37  90 

31  39 

27  06 

26 

20  68 

50  04 

37  00 

30  63 

26 

101  94 

65  89 

48  26 

88  04 

31  56 

27  26 

27 

21  15 

50  87 

87  63 

31  16 

27 

102  04 

66  00 

48  39 

38  19 

81  73 

27  49 

28 

21  69 

51  72 

38  27 

81  71 

28 

102  14 

66  11 

48  52 

88  85 

31  93 

27  73 

29 

22  26 

52  61 

88  94 

32  28 

29 

102  25 

66  24 

48  67 

38  52 

32  14 

28  00 

30 

22  85 

53  52 

89  64 

82  87 

30 

102  37 

66  37 

48  83 

88  71 

82  88 

28  29 

31 

23  48 

5446 

40  86 

33  49 

31 

102  49 

66  52 

49  00 

8892 

82  68 

28  61 

32 

24  14 

55  44 

41  10 

34  13 

32 

102  68 

66  68 

49  18 

89  14 

82  92 

28  96 

33 

24  84 

56  45 

41  88 

34  80 

33 

102  77 

66  85 

49  38 

39  39 

83  23 

29  35 

34 

25  58 

57  50 

42  68 

35  49 

34 

102  93 

67  03 

49  60 

89  67 

83  57 

29  78 

35 

26  85 

58  58 

43  51 

36  22 

35 

103  10 

67  23 

49  85 

89  97 

83  95 

30  24 

36 

27  17 

59  70 

44  88 

36  98 

86 

108  28 

67  45 

50  11 

40  80 

34  86 

30  76 

37 

28  04 

60  86 

45  28 

37  77 

37 

103  48 

67  68 

50  41 

40  67 

84  82 

31  88 

38 

28  95 

62  06 

46  22 

38  60 

38 

103  69 

67  94 

50  73 

41  07 

35  33 

31  95 

39 

29  92 

63  80 

47  20 

39  47 

39 

103  93 

68  23 

51  09 

41  52 

85  Mi 

32  68 

40 

80  94 

64  59 

48  22 

40  38 

40 

104  18 

68  55 

51  48 

42  02 

36  50 

33  88 

41 

82  03 

65  93 

49  28 

41  34 

41 

104  46 

68  90 

51  92 

42  57 

37  18 

34  20 

42 

88  18 

67  31 

50  89 

42  35 

42 

104  77 

69  28 

52  41 

48  17 

87  93 

35  10 

43 

84  40 

68  76 

51  56 

43  41 

43 

105  11 

69  71 

52  95 

43  85 

38  76 

:;c,  (is 

44 

85  70 

70  25 

52  78 

44  54 

44 

105  49 

70  19 

58  55 

44  59 

39  67 

37  15 

45 

87  08 

71  81 

54  06 

45  73 

45 

105  92 

70  73 

54  22 

45  42 

40  67 

38  82 

46 

88  55 

73  44 

55  40 

4699 

46 

10689 

71  32 

54  96 

46  83 

41  78 

47 

40  12 

75  13 

56  82 

48  88 

47 

106  91 

71  98 

55  78 

47  84 

42  99 

48 

41  78 

76  90 

58  31 

49  75 

48 

107  50 

72  71 

5fi  69 

48  46 

44  81 

49 

43  56 

78  74 

59  88 

51  26 

49 

108  15 

73  53 

57  70 

49  69 

45  76 

.  . 

50 

45  45 

80  66 

61  54 

52  87 

50 

108  87 

74  43 

58  81 

51  05 

47  85 

847.  The  following  tables  illustrate  the  three  options  of  the  in- 
sured if  he  ceases  to  pay  premiums  before  the  maturity  of  the  policy  : 
(1)  to  receive  a  certain  amount  of  cash  at  once  j  or  (2)  to  be  insured 


§§847-848] 


INSURANCE 


323 


for  the  amount  of  the  policy  in  case  of  death  within  a  limited  time ; 
or  (3)  to  be  insured  for  a  certain  smaller  amount  in  case  of  death 
at  any  time. 


SPECIMEN  TABLE   INDORSED  ON 
ORDINARY  LIFE  POLICIES  FOR 

$1000 


AGE,  35 


PREMIUM,  $26.35 


SPECIMEN  TABLE  INDORSED  ON  20- YEAR 
ENDOWMENT  POLICIES  FOR  $10,000 


AGE,  35 


PREMIUM,  $498.50 


AUTOMATIC 

AT  END 

CASH 
SURRENDER 

EXTENDED 
INSURANCE 

PAID-UP 

AT  END 

CASH 
SURRENDER 

AUTOMATIC  EX- 
TENDED INSURANCE 

PAID-UP 

OF 

VALUE 

POLICY 

U1T 

VALUE 

POLICY 

YEAR 

LOAN  VALUE 

Years 

Days 

YEAR 

LOAN  VALUE 

Years 

Days 

Pure  En- 
dowment 

2d 

$16.13 

1 

297 

$37.00 

2d 

$604.00 

7 

46 

$980.00 

3d 

29.76 

3 

122 

67.00 

3d 

975.00 

11 

185 

1540.00 

4th 

43.77 

4 

313 

97.00 

4th 

1359.10 

15 

203 

2090.00 

5th 

58.16 

6 

132 

127.00 

5th 

1757.10 

15 

$750.00 

2640.00 

6th 

72.94 

7 

292 

156.00 

6th 

2169.80 

14 

1550.00 

3170.00 

7th 

88.11 

9 

47 

185.00 

7th 

2596.60 

13 

2320.00 

8710.00 

8th 

103.68 

10 

115 

213.00 

8th 

3039.40 

12 

3060.00 

4230.00 

9th 

119.65 

11 

128 

242.00 

9th 

3498.50 

11 

8760.00 

4740.00 

10th 

136.01 

12 

86 

270.00 

10th 

3974.50 

10 

4450.00 

5250.00 

llth 

152.76 

12 

357 

297.00 

llth 

4468.40 

9 

5100.00 

5750.00 

12th 

169.87 

13 

215 

324.00 

12th 

4980.80 

8 

5730.00 

6240.00 

13th 

187.35 

14 

32 

351.00 

13th 

5512.80 

7 

6330.00 

6720.00 

14th 

205.16 

14 

175 

377.00 

14th 

6065.50 

6 

6910.00 

7200.00 

15th 

223.28 

14 

284 

402.00 

15th 

6640.00 

5 

7460.00 

7660.00 

20th 

317.58 

15 

86 

521.00 

16th 

7237.70 

4 

7990.00 

8120.00 

25th 

415.49 

14 

215 

623.00 

17th 

7860.50 

3 

8500.00 

8580.00 

30th 

512.92 

13 

149 

709.00 

18th 

8510.10 

2 

8990.00 

9020.00 

35th 

605.14 

11 

326 

779.00 

19th 

9189.10 

1 

9460.00 

9460.00 

40th 

688.21 

10 

48 

834.00 

20th 

10000.00 

10000.00 

10000.00 

848.  Two  important  special  kinds  of  insurance  are  somewhat 
similar  in  character  to  personal  insurance  : 

1.  Guaranty  and  fidelity  insurance  is  indemnity  for  loss  because  of 
fraud,  dishonesty,  or  negligence  on  the  part  of  agents  or  employees. 

2.  Bonding  companies  guarantee  the   payment   of    bonds   given 
by  an  agent,  contractor,  treasurer,  secretary,  or  employee,  for  the 
proper  performance  of  some  specific  or  general  line  of  duty. 

For  example,  when  the  New  York  subway  was  contracted  for,  the  con- 
tractor was  obliged  to  give  the  city  of  New  York  bonds  for  some  $  35,000,000. 
A  number  of  bonding  companies  agreed  that  if  the  contractor  failed  entirely  01 
in  part  to  execute  his  contract,  these  bonding  companies  would  indemnify  the 
city  of  New  York  for  the  loss  caused  by  his  failure. 

Bonding  companies  have  certain  rates  for  the  risk  they  assume.  For  build- 
ing tunnels  the  charge  is  usually  1%  on  the  amount  of  contract  up  to  $60,000, 
and  \  %  for  any  excess.  For  instance,  if  George  L.  Benton  makes  a  contract  to 


324  PERCENTAGE   AND   ITS  APPLICATIONS         [§§848-849 

build  a  tunnel  for  $  65,000,  the  cost  of  a  bond  would  be  1  %  on  $  50,000,  or  $  500, 
and  £%  on  $  15,000,  or  $75  ;  a  total  of  $575.  The  usual  charge  for  bonds  for 
agents  working  on  a  salary  is  60  ?  per  hundred  dollars  up  to  $  2500,  and  50  ^  per 
hundred  for  any  excess  over  $  2500  ;  minimum  charge,  $  7.50.  For  instance,  if  an 
agent  were  obliged  to  give  a  bond  of  $  4000,  the  charge  would  be  $  22.50  (60  ^  per 
hundred  on  $2500,  or  $  15,  and  50  j*  per  hundred  on  $  1500,  or  $  7.50).  Should 
the  bond  be  for  either  $  500  or  $  1000,  the  minimum  charge  of  $  7.50  would  apply. 

849.  Examples  in  life  insurance.  1.  What  continuous  premium 
should  be  paid  annually  to  secure  $7500  at  death,  if  the  insured 
is  39  years  of  age  at  the  time  the  policy  is  issued  ? 

SOLUTION 

The  table,  p.  322,  shows  for  age  39,  continuous  premiums  of  $29.92  per  $1000. 
$29.92  x  7.5  =  $224.40,  the  annual  premium  for  $7500. 

2.  A,  42  years  of  age,  wishes  to  secure  a  policy  of  $  15,000  pay- 
able at  his  death.     How  much  premium  would  he  be  obliged  to  pay 
annually  if  he  chose  to  have  a  paid-up  policy  after  making  ten  pre- 
mium payments  ? 

SOLUTION 

In  the  10-premiums  column,  opposite  age  42,  we  find  $67.31. 

$67.31  x  15  =  $1009.65,  the  amount  of  each  of  the  ten  annual  premiums. 

3.  B  is  27  years  of  age  and  secures  a  20-year  endowment  policy 
for  $20,000.     How  much  is  he  obliged  to  pay  at  the  beginning  of 

each  year  ? 

SOLUTION 

In  the  table,  p.  822,  opposite  age  27,  we  find  in  the  20-years  column,  $48.39. 
$48.39  x  20  =  $967.80,  one  of  the  twenty  premiums. 

4.  If  B  had  lived  to  be  47  years  of  age  and  had  put  the  amount 
of  the  premium  in  example  3  in  a  savings  bank  at  the  beginning  of 
each  year,  at  3  %  compound  interest,  how  much  would  be  due  him  ? 
On  this  basis,  what  was  the  amount  of  the  total  cost  of  the  pure  in- 
surance (insurance  for  the  difference  between  cash  surrender  value 
and  face  of  policy)  in  example  3  ? 

SOLUTION 

$1  placed  at  3%  compound  interest  at  the  beginning  of  each  year  amounts  in 
20  years  to  $27.677.  (See  table,  p.  398.) 

$27.677  x  967.80  =  $26,785.80,  the  amount  due  from  the  savings  bank. 

$26,785.80  -  $20,000  (cash  value  of  policy  at  age  47)  =  $6785.80,  the  amount 
of  the  cost  of  the  pure  insurance. 


§  849]  INSURANCE  325 

WRITTEN  EXERCISE 

1.  What  premium  must  be  paid  annually  upon  a  life  policy  to  give 
my  beneficiary  $6000  upon  my  decease,  if  I  am  now  35  years  of  age  '/ 

2.  X  takes  out  an  ordinary  life  policy  for  $  10,000  in  favor  of  his 
wife.     What  are  his  annual  premiums  if  he  is  now  37  years  of  age  ? 

3.  Y  is  32  years  old  and  takes  out  a  15-year  endowment  policy  for 
$5000.    What  premium  must  he  pay  at  the  beginning  of  each  year? 

4.  In  example  3,  if  Y  had  lived  to  be  47  years  of  age  and  had 
put  the  amount  of  the  yearly  premium  in  a  bank  each  year,  how 
much  would  be  due  him  at  3  %  compound  interest  ?     How  much 
more  would  the  bank  then  pay  than  the  insurance  company  ? 

5.  A  life  insurance  company  issued  a  10-year  endowment  policy 
for  $  .t2,000  to  a  man  32  years  of  age.     The  insured  died  at  38 
years  of  age.     How  much  more  than  the  face  of  the  premiums  did 
the  company  have  to  pay?     If  the  company  invested  each  year  as  a 
reserve  fund  60  %  of  the  annual  premium  at  4  %  compound  interest 
(see  p.  398),  how  much  more  or  less  than  the  premiums  and  the 
interest  they  earned  did  the  company  pay  ? 

6.  A  man  35  years  of  age  takes  out  an  ordinary  life  policy  for 
$  5000.     At  the  end  of  the  tenth  year  he  exercises  his  loan  privilege 
and  borrows  from  the  company  the  cash  surrender  value  of  the 
policy.     How  much  does  he  get  ?    If  he  had  deposited  the  annual 
premiums  in  a  savings  bank  paying  3  %  compound  interest,  how 
much  could  he  have  withdrawn  from  the  bank  after  ten  years  ? 

7.  Two  persons,  each  38  years  of  age,  insured  for  $  30,000  each. 
One  secured  a  life  policy  and  the  other  a  10-year  endowment  policy. 
How  much  had  each  paid  in  premiums  in  eight  years? 

8.  A  man  32  years  of  age  secured  a  15-year  endowment  policy 
for  $  8000.     If  he  died  at  the  end  of  the  seventh  year,  how  much 
more  than  the  premiums  paid  in  did  the  insurance  company  have  to 
pay  his  beneficiary  ?     If  the  company  loaned  50  %  of  the  premium 
at  the  beginning  of  each  year  at  3|  %  compound  interest,  how  much 
did  it  lose  on  the  policy  ? 

9.  Y  obtained  a  20-year  endowment  policy  at  the  age  of  28  years. 
How  much  more  or  less  than  the  face  value  of  the  policy  will  he  pay 
in  per  thousand  dollars  in  actual  premiums  if  he  survives  the  twenty 


326  PERCENTAGE    AND   ITS   APPLICATIONS          [§§849-855 

years?     At  3  %  simple  interest  what  would  these  premiums  have 
yielded  in  principal  and  interest  in  twenty  years  ? 

10.  M  insured  his  life  at  the  age  of  41  years  for  $  25,000  and 
paid  the  annual  ordinary  premiums  till  his  death  at  76  years  of  age. 
How  much  did  he  pay  in  premiums  ? 

11.  A,  aged  35  years,  and  O,  aged  33  years,  two  partners,  each 
insured  his  life   in  favor   of  the   other  by  an  ordinary  policy  for 
$  5000.     At  the  end  of  five  years  0  died.     A  now  receives  the  face 
of  O's  policy,  and  also  exercises  his  right  to  take  the  cash  surrender 
value  of  his  own  policy.     How  much  more  than  the  premiums  paid  in 
by  both  men  does  A  get  ?     If  the  amount  of  the  two  premiums  had 
been  placed  at  compound  interest  each  year  at  3  %,  how  much  more 
or  less  than  the  sum  received  by  A  would  they  have  amounted  to? 

12.  What  is  the  cash  surrender  or  loan  value  for  an  oidinary 
$  5000  policy  issued  to  a  man  aged  35  years,  after  he  has  paid  seven 
years'  premiums?     If  he  takes  extended  insurance  instead  of  cash 
surrender  value,  how  long  will  it  extend?     What  would  be  the 
amount  of  a  paid-up  policy  if  he  chooses  that  ? 

IS.  What  would  be  the  answer  to  the  three  questions  in  Ex.  12 
if  the  policy  were  a  20-year  endowment  policy  for  $  10,000  ? 

TAXES 

850.  A  tax  is  a  sum  of  money  levied  on  the  person,  property,  busi- 
ness, or  income  of  an  individual  for  the  support  of  the  government 
or  for  any  public  purpose. 

851.  The  taxes  levied  by  the  national  government  are  indirect 
taxes;  they  consist  of  customs  duties  and  internal  revenue. 

852.  The  taxes  levied  by  the  state  and  local  governments  are 
mostly  direct  taxes;  they  consist  of  poll  tax  and  property  tax. 

853.  A  poll  tax  is  a  tax  levied  on  a  person  without  regard  to  the 
property  he  owns.    It  is  a  certain  amount  per  each  adult  male  citizen. 

854.  A  property  tax  is  a  tax  assessed  upon  property  at  a  given 
rate  per  cent  of  the  valuation. 

855.  Property  is  of  two  kinds :  personal  and  real. 


§§  856-862]  TAXES  327 

856.  Personal  property  is   any  movable  property,  such  as  mer- 
chandise, ships,  cattle,  money,  stocks,  mortgages. 

857.  Real  property,  or  real  estate,  is  any  fixed  or  immovable  prop- 
erty, such  as  houses,  lands,  mines,  quarries. 

858.  An  assessor  is  a  public  or  government  officer,  appointed 
to  estimate  the  value  of  property  to  be  taxed,  and  to  apportion  the 
taxes  in  proportion  to  the  value  of  each  man's  property. 

859.  An  assessment  roll  is  a  descriptive  list  of  taxable  property. 
It  shows  in  some  detail  the  names  of  the  owners  of  the  property  in 
the  district  assessed,  its  location,  and  assessed  valuation. 

860.  A  collector  is  a  public  officer  appointed  to  receive  and  collect 
taxes. 

Taxes  are  generally  assessed  and  made  payable  in  money,  but  road  taxes 
are  sometimes  made  payable  in  day's  work. 

The  methods  of  collecting  taxes  vary  in  different  states.  In  some  states  all 
the  taxes  are  collected  in  the  several  counties,  while  in  others  they  are  all  col- 
lected in  the  several  towns.  In  almost  all  the  states  the  different  taxes, — 
state,  county,  town,  school,  etc.,  — are  aggregated  ;  that  is,  paid  in  one  amount. 

In  certain  states  the  common  schools  are  supported  by  a  tax  or  a  rate  bill, 
made  out  on  a  basis  of  the  total  attendance. 

861.  The  rate  of  taxation  is  the  sum  charged  on  each  dollar  of 
the  assessed  valuation  to  raise  the  required  amount  of  taxes. 

862.  Computations   in  taxes  are  made  in  accordance  with  the 
general  principles  of  abstract  percentage,  the  assessed  valuation  of 
the  property  corresponding  to  the  base;  the  rate  of  taxation,  to  the 
rate ;  and  the  tax  to  the  percentage. 

DRILL  EXERCISE 

1.  If  the  rate  of  taxation  is  70  cents  on  each  $  100,  how  much 
tax   must  I  pay  on  property,  the   assessed  valuation  of  which  is 
$  9000  ? 

2.  If  a  man  pays  $  600  tax  on  property  worth  $120,000,  what  is 
the  rate  of  taxation  ? 

8.  I  have  property  assessed  at  $  12,000  and  pay  /or  2  polls  at 
$  2.50  each.  If  my  total  tax  is  $  65,  how  many  cents  on  the  dollar 
is  the  tax  rate  ? 


328  PERCENTAGE   AND  ITS  APPLICATIONS         [§§  862-867 

4-   G-iven  the  valuation  and  rate  of  taxation,  how  may  the  tax  be 
found? 

5.  Given  the  tax  and  the  rate  of  taxation,  how  may  the  assessed 
valuation  be  found  ? 

6.  Given  the  assessed  valuation  and  the  tax  to  be  raised,  how 
may  the  rate  of  taxation  be  found  ? 

863.  To  find  a  property  tax. 

WRITTEN  EXERCISE 

1.  Henry  Wilson  is  assessed  $4000  on  his  real  estate  and  $3500 
on  his  personal  property.     If  the  rate  of  taxation  is  2.5  mills  on  $1, 
what  is  the  amount  of  his  tax  ? 

2.  The  taxable  property  of  a  town  is  $  472,500,  and  the  rate  of 
taxation  is  2.4  mills  on  $  1.     What  is  the  amount  of  tax  to  be  raised, 
and  how  much  should  B  pay,  who  is  assessed  $4000  on  his  real 
estate  and  $  1600  on  his  personal  property  ? 

864.  To  find  a  general  tax. 

865.  Example.     A  tax  of  $  2505  is  to  be  assessed  upon  a  certain 
village.      The  valuation  of  the  taxable  property  is  $600,000   and 
there  are  324  polls  to  be  assessed  at  $  1.25  each.     What  will  be  the 
tax  on  $  1,  and  how  much  will  be  the  tax  of  Mr.  Scott  whose  prop- 
erty is  valued  at  $  12,500  and  who  pays  for  2  polls  ? 

SOLUTION 

$1.25  x  324  =  $405,  the  amount  of  poll  tax. 
$2505  -  $  405  =  $  2100,  the  amount  of  property  tax. 
$2100  -4-  $600,000  =  .0035,  the  rate  of  taxation. 
$  12,500  x  .0035  =  $43.75,  Mr.  Scott's  property  tax. 
$1.25  x  2  =  $2.50,  Mr.  Scott's  poll  tax. 
$43.75  +  $2.50  =  $46.25,  Mr.  Scott's  total  tax. 

866.  To  find  the  rate  of  taxation. 

867.  Example.     A  tax  of  $  3750  is  to  be  levied  upon  a  valuation 
of  $1,250,000.     Find  the  rate  of  taxation. 

SOLUTION 

$3750  +  $  1,250,000  =  .003. 

Therefore,  the  rate  of  taxation  is  $  .003  on  $1. 


§  867]  TAXES  329 

WRITTEN  EXERCISE 

1.  A  tax  of  $37,500  is  levied  on  a  city,  the  assessed  valuation 
of  which  is  $2,500,000.     What  is  the  rate  of  taxation,  and  what 
amount  of  tax  will  a  person  have  to  pay  who  is  assessed  $  4500  on 
his  real  estate  and  $  3500  on  his  personal  property  ? 

2.  The  cost  of  a  new  schoolhouse  was  $3500.     If  the  taxable 
property  of  the  district  is  assessed  at  $  700,000,  what  is  the  tax  rate 
on  $  100  ?    What  is  B's  tax  if  he  is  assessed  $  250  on  his  real  estate 
and  $  2000  on  his  personal  property  ? 

WRITTEN  REVIEW 

1.  A  tax  of  $125,000  is  levied  on  a  city,  the  assessed  valuation 
of  which  is  $15,000,000.  What  is  the  rate  of  taxation,  and  what 
amount  of  tax  will  a  person  have  to  pay  whose  property  is  valued 
at  $7500? 

8.  If  a  tax  of  $120  is  assessed  on  a  mill  valued  at  $24,000, 
what  is  the  valuation  of  a  residence  that  is  taxed  $17.75  at  the 
same  rate? 

8.  The  tax  assessed  upon  a  town  is  $20,914.80;  the  town  con- 
tains 2580  polls,  taxed  $  .62^  each,  and  has  a  real  estate  valuation 
of  $  4,062,000,  a*nd  a  valuation  of  personal  property  to  the  amount 
of  $227,400.  Find  the  rate  of  taxation,  and  C's  tax,  who  pays  for 
4  polls,  and  whose  property  is  assessed  at  $15,000. 

^.  My  son  and  daughter  each  attended  school  214  days,  and  the 
expense,  including  teacher's  wages  and  incidentals,  was  paid  by  a 
rate  bill.  How  much  must  I  pay  if  the  teacher's  wages  amounted 
to  $440,  fuel  and  repairs  $101.50,  and  janitor's  fees  $74.75,  the 
total  number  of  days'  attendance  being  7460  ? 

5.  A  tax  of  $  24,000  is  levied  upon  a  town  which  has  taxable 
property  with  the  assessed  valuation  of  $  1,500,200.  and  which  con- 
tains 520  polls,  assessed  at  $1.25  each.  The  town  receives  from  the 
state  $  4000  as  its  share  of  the  corporation  taxes.  Find  the  rate  of 
taxation,  and  the  amount  of  tax  to  be  paid  by  James  Brown,  who  is 
assessed  $4200  on  his  real  estate  and  $1800  on  his  personal  prop- 
erty, and  who  pays  for  4  polls. 


330  PERCENTAGE   AND  ITS   APPLICATIONS  [§  867 

6.  The  cost  of  maintaining  the  public  schools  of  a  city  during 
the  year  1903  was  $  112,000,  and  the  taxable  property  of  the  city 
was  $  44,800,000.  How  many  mills  on  a  dollar  must  be  assessed  for 
school  purposes  ?  If  10  %  of  the  tax  cannot  be  collected,  how  many 
mills  on  a  dollar  must  then  be  assessed  ? 

7.  A  tax  of  $  13,943.20  is  assessed  upon  a  town  containing  860 
taxable  polls ;  the  real  estate  is  valued  at  $  2,708,000,  and  the  per- 
sonal property  at  $  151,600.     If  the  polls  be  taxed  $  1.25  each,  what 
will  be  the  rate  of  property  taxation,  and  what  will  be  the  tax  of 
Frederick  Benton,  who  pays  for  3  polls,  and  has  real  and  personal 
estate  valued  at  $  23,750? 

8.  In  a  school  district  the  valuation  of  the  taxable  property 
is  $  752,400,  and  it  is  proposed  to  repair  the  schoolhouse  and  orna- 
ment the  grounds  at  an  expense  of  $  5000.     If  old  materials  sell  for 
$673.70,  what  will  be  the  rate  per  cent  of  taxation,  and  what  will  be 
A's  tax,  whose  property  was  valued  at  $  9400  ? 

9.  The   assessed  valuation  of  the  real   estate  of  a  county  is 
$1,910,887,  of  the   personal   property,  $921,073,  and   it  has  4564 
inhabitants  subject  to  a  poll  tax.     The  year's  expenses  are :    for 
schools,  $  8400 ;  interest,  $  6850 ;  highways,  $  7560 ;  salaries,  $  5150 ; 
and  contingent   expenses,   $13,675.      If  the  poll  tax  was  $1.50, 
and  the  revenue  from  fairs  and   licenses   $  6200,  what   must    be 
levied  on  a  dollar  to  meet  expenses  and  provide  a  sinking  fund  of 
$7000? 

10.  In  a  certain  town  there  are  680  polls.  The  assessed  valua- 
tion of  the  real  estate  is  $850,000,  and  of  the  personal  property 
$  750,000.  The  poll  tax  is  $  1*50  per  poll,  and  the  tax  on  the  property 
is  1|%.  Only  98%  of  the  whole  tax  can  be  collected,  and  the 
collector  is  paid  2|%  of  the  amount  collected.  How  much  does  the 
town  receive  from  the  tax,,  and  how  much  does  the  collector  receive 
for  his  services? 

NOTE.  It  is  suggested  that  the  teacher  give  additional  examples  in  taxes 
according  to  the  regulations  of  his  own  town  or  city.  These  regulations  can 
generally  be  obtained  with  little  trouble  from  the  local  tax  collector  or  other 
officers  of  the  local  government.  The  work  on  the  next  three  pages  lias  been 
selected  to  give  practice,  if  desired,  in  examples  under  regulations  of  more  than 
usual  intricacy. 


§868] 


TAXES 


331 


868.  The  following  regulations  as  to  assessments  and  taxes  are 
those  for  Philadelphia  under  the  laws  of  the  state  of  Pennsylvania: 

DISCOUNT  ON  CITY  TAXES 

One  per  cent  discount  on  Bills  paid  during 
January,  February,  and  March, 

After  the  last  day  of  March  a  dis- 
count at  the  rate  of  i  per  cent  per 
annum  on  all  bills  paid  ON  OR  BE- 
FORE JUNE  30.  Beginning  April 
1  with  |  of  one  per  cent,  and  de- 
creasing daily  until  June  30,  when 
discount  will  be  |  of  one  per  cent. 
No  discount  or  penalties  on  bills  paid 
during  July  and  August. 

PENALTIES  ON  CITY  TAXES 

After  August  31,  1  per  cent. 
After  September  30,  2  per  cent. 
After  October  31,  3  per  cent. 
After  November  30,  4  per  cent. 

After  December  31,  Delinquent, 

No  Discount  is  allowed  on  State 
Tax,  but  a  Penalty  of  5  per  cent  is 
added  after  July  31st. 

Every  person  twenty-one  years  of  age  and  upwards,  being  a  .resident  of  or 
domiciled  within  this  State,  and  every  corporation  not  specially  exempted,  and 
every  co-partnership  or  unincorporated  association,  joint  stock  association  or  com- 
pany, limited  partnership,  and  co-partnership,  located  or  doing  business  within  this 
Commonwealth,  owning  or  holding  any  personal  property  of  the  classes  enumerated 
in  Section  1  of  the  Act  of  June  8,  1891  [consisting  of  horses,  cattle,  vehicles  to  hire, 
and  money  at  interest],  whether  the  same  be  held  in  his,  her,  or  its  own  right  or  as 
Trustee,  Executor,  Administrator,  Guardian,  Assignee,  Committee,  Receiver,  or  in 
any  other  Fiduciary  capacity  for  the  use  and  benefit  of  some  other  person  or  corpora- 
tion, is  required  each  year  to  make  return,  under  oath,  of  the  amount  of  such  prop- 
erty, to  the  Assessor. 

In  case  no  return  is  made  within  ten  days  the  assessors  are  required  to  make  one, 
to  which  estimated  return  50  per  cent  is  to  be  added  subject  to  appeal  as  provided  by 
law. 


TAX    BATES 

I9O7 

$100 
100 
100 

100 

Mills 
Mills 

)WS 
5 
23 
9 
30 
7 
10 
18 
5 
7 
2 
6 
2 
26 

Real    Estate,  full    city    rate, 
<$  i  50  ^    .     .         .... 

Real  Estate,  suburban,  $1.00  ^ 
Real  Estate,  farm,  $  .75  $ 
Horses,     Mules,    and    Cattle, 

Vehicles  for  Hire,  State  Tax,  4 
Money  at  Interest,  State  Tax,  4 

APPOKTIONED  AS  FOLLC 

Poor,    

Lighting  City    

Loan                   ...... 

Highways,     
Water,      

Police,  

Markets  and  City  Property,    . 
Fire      

Prisons,     

City  Commissioners,  .... 
Health,     
Expense  of  Municipality,  .     . 

$1.50 

332 


PERCENTAGE   AND   ITS  APPLICATIONS 


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§§869-870]  CUSTOMHOUSE  BUSINESS  333 

869.  In  the  assessment  book,  p.  332,  the  full  city  rate  of  $  1.50  per 
$  100  applies  to  the  improved  real  estate  when  not  otherwise  specified. 
The  suburban  rate  (see  table,  p.  331)  is  levied  on  "  unimproved  prop- 
erty " — chiefly    areas    relatively    small    fronting    on    unimproved 
streets.     The  farm  rate  is  levied  on  tracts  of  relatively  large  area 
used  for  farming  and  usually  lying  in  the  rear  of  lands  fronting 
upon  some  improved  street.     The  state  tax  of  4  mills  per  dollar  on 
vehicles  to  hire  and  money  at  interest  is  collected  by  the  city  and 
sent  to  the  state  j  but  the  state  then  returns  f  of  it  to  the  city. 

WRITTEN  EXERCISE 

1.  From  the  table  of  rates  and  the  assessment  book,  pp.  331,  332, 
how  much  tax  is  levied  on  the  property  1400-1408  Chestnut  St.  ? 

2.  What  is  Thomas  Dolan's  total  tax?    If  he  had  failed  to  make 
return  for  the  value  of  his  property,  how  much  would  his  tax  have 
been  ?     (See  last  paragraph  on  p.  331.) 

3.  What  is  the  total  tax  of  the  persons  listed  on  p.  332  ? 

4.  How  much  of  this  total  tax  would  the  state  finally  keep  ? 

5.  If  the  trustees  of  the  William  Blanchard  estate  did  not  pay 
the  real  estate  tax  till  Dec.  5,  how  much  would  the  penalty  be  ? 

6.  What  is  the  penalty  if  Anna  Blanchard  does  not  pay  her  tax 
till  Aug.  5  ? 

7.  If  the  Land  Title  Trust  Co.  paid  their  real  estate  tax  in 
March,  what  was  the  discount  ?     How  much  did  they  have  to  pay  ? 

8.  If  the  Corporation  of  Haverford  College  paid  their  city  tax 
March  20,  their  suburban  tax  March  30,  and  their  farm  tax  July  5, 
what  was  the  total  discount  ? 

NOTE.  The  teacher  can  easily  vary  the  number  of  examples  as  desired,  "by 
replacing  example  3  with  a  few  specific  examples,  and  by  giving  additional 
examples  based  on  the  regulations  for  discounts,  penalties,  etc. 

CUSTOMHOUSE  BUSINESS 

870.  A  customhouse  is   an   office    established    by  the   national 
government   for  the  transaction  of  business  relating  to  duties,  or 
customs,  and  for  the  entry  and  clearance  of  vessels. 


334  PERCENTAGE   AND  ITS  APPLICATIONS          [§§  871-876 

871.  A  port  of  entry  is  a  port  at  which  a  customhouse  is  estab- 
lished for  the  legal  entrance  of  vessels  and  merchandise. 

The  waters  and  shores  of  the  United  States  are  divided  into  collection  dis- 
tricts in  each  of  which  there  is  a  port  of  entry  which  is  also  a  port  of  delivery  ; 
other  ports  than  those  of  entry  may  be  specified  as  ports  of  delivery.  Duties  are 
paid  (or  secured  to  be  paid),  and  clearances  made,  at  ports  of  entry  only,  but 
after  vessels  have  been  properly  entered,  their  cargoes  may  be  discharged  at  any 
port  of -delivery. 

872.  Duties,  or  customs,  are  taxes  levied  by  the  national  govern- 
ment upon  imported  goods.     They  are  of  two  kinds,  ad  valorem  and 
specific. 

873.  An  ad  valorem  duty  is  a  certain  per  cent  levied  on  the 
appraised  value  of  the  goods,  which  is  the  market  value  in  the 
country  from  which  they  are  imported. 

Ad  valorem  duties  are  not  computed  on  fractions  of  a  dollar;  if  the  cents  in 
an  invoice  are  less  than  50,  they  are  rejected ;  if  50  or  more,  they  are  counted 
as  another  dollar.  On  pages  338-340  it  is  assumed  (unless  otherwise  stated)  that 
the  appraised  value  corresponds  to  the  invoiced  cost. 

874.  A  specific  duty  is  a  tax  levied  upon  the  number,  weight,  or 
measure  of  goods,  regardless  of  their  value ;  as,  a  fixed  sum  per  bale, 
ton,  barrel,  etc. 

Upon  some  goods  both  specific  and  ad  valorem  duties  are  levied.  Before 
specific  duties  are  finally  determined,  allowances  are  made  for  tare,  leakage,  etc. 

875.  An  invoice,  or  manifest,  is  a  written  account  of  the  particular 
goods  sent  to  the  purchaser  or  consignee,  showing  the  quantity  and 
the  actual  cost  or  value  of  the  goods. 

All  invoices  must  be  made  out  in  the  weights  and  measures  of  the  place 
or  country  from  which  the  goods  are  imported,  and  in  the  currency  of  that 
country  or  in  the  currency  actually  paid  for  them. 

When  the  value  of  foreign  currency  is  fixed  by  law,  such  value  must  be 
taken  in  estimating  the  duties. 

876.  The  value  in  United  States  money  of  the  foreign  currency 
of  the  different  nations  of  the  world  is  proclaimed  by  the  Secretary 
of  the  Treasury  every  three  months.     The  following  values  of  for- 
eign coins  were  proclaimed  Apr.  1,  1907 : 


§§  87C-880] 


CUSTOMHOUSE   BUSINESS 
VALUES  OF  FOREIGN  COINS 


335 


COUNTRY 

STANDARD 

MONETARY  UNIT 

VALUE  IN 
U.  S.  GOLD 

Argentina          .... 
Austria-Hungary 

Gold 
Gold 
Gold 

Peso 
Crown 
Peso 

$.965 
.203 
.9733 

Brazil        ..... 
Chile         
Denmark,  Norway,  Sweden     . 
Egypt        
France,  Belgium,  Switzerland  . 
German  Empire 
Great  Britain,  India 
Japan        ..... 

Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 

Milreis 
Peso 
Crown 
Pound 
Franc 
Mark 
Pound  sterling 
Yen 
Peso 

.546 
.365 
.268 
4.943 
.193 
.238 
4.8665 
.498 
.498 

Netherlands      .... 
Newfoundland  .... 
Philippine  Islands    . 

Gold 
Gold 
Gold 
Gold 

Florin 
Dollar 
Peso 
Milreis 

.402 
1.014 
.50 
1.08 

Russia 

Gold 

Ruble 

.515 

Turkey      

Gold 
Gold 

Piaster 
Peso 

.044 
1  034 

The  drachma  of  Greece,  the  lira  of  Italy,  the  peseta  of  Spain,  the  bolivar  of 
Venezuela  are  of  the  same  value  as  the  franc.  The  dollar,  of  the  same  value  as 
our  own,  is  the  standard  of  the  British  possessions  of  North  America  (except 
Newfoundland),  of  Liberia,  and  of  Colombia.  The  libra  of  Peru  has  the  same 
value  as  the  British  pound  sterling.  The  gourde  of  Haiti  has  the  same  value  as 
the  peso  of  Argentina. 

877.  A  tariff  is  a  schedule  of  goods,  and  the  legal  rates  of  import 
duties  imposed  by  law  on  the  same. 

878.  A  free  list  is  a  list  of  such  articles  as  are  exempt  from  duty. 

879.  Tonnage  is  a  tax  levied  upon  a  vessel,  independent  of  its 
cargo,  for  the  privilege  of  coming  into  a  port  of  entry. 

880.  Duties  are  collected  at  the  port   of   entry  by  a  customs 
officer  appointed  by  the  United  States  government,  and  known  as 
the  Collector  of  the  Port.     Under  him  are  deputy  collectors,  surveyors, 
and  appraisers,  and  many  inspectors,  weighers,  gangers,  etc. 


336  PERCENTAGE  AND   ITS  APPLICATIONS         [§§  881-888 

881.  A  naval  officer,  appointed  only  at  the  more  important  ports, 
receives  copies  of  all  manifests,  countersigns  all  documents  issued 
by  the  collector,  and  certifies  his  estimates  and  accounts. 

882.  The  surveyor  is  the  outdoor  executive  officer  of  the  port. 
He  supervises  the  inspectors,  controls  the  unlading  of  foreign  mer- 
chandise, etc. 

883.  The  appraiser  examines   imported    merchandise   and   de- 
termines its  dutiable  value;   that  is,  the  foreign  market  value  at 
the  time  of  exportation. 

884.  The  public  store  is  a  place  provided  for  the  examination 
of  imported  merchandise. 

One  package  of  every  invoice  of  merchandise,  and  at  least  one  package  out 
of  every  ten  similar  packages,  must  be  sent  to  the  public  store  for  examination. 

Bulky  and  heavy  articles  are  examined  at  the  wharf  where  they  are  unloaded. 

Weighable  and  gaugeable  goods  paying  only  specific  duties  are  seldom  sent 
to  the  public  store  for  examination. 

885.  Warehousing  is  the  depositing  of  imported  goods  in  a  gov- 
ernment or  bonded  warehouse. 

886.  A  bonded  warehouse  is  a  place  provided  by  law  for  the 

storage  of  dutiable  merchandise. 

Goods  may  be  withdrawn  from  a  bonded  warehouse  for  export  without  the 
payment  of  the  duties.  If  goods  on  which  the  duty,  amounting  to  $50  or  more, 
has  been  paid  are  exported,  the  amount  of  duty,  less  1  %,  is  refunded  ;  the  sum 
so  refunded  is  called  a  drawback. 

887.  Smuggling  is  the  act  of  bringing  foreign  goods  into  a  coun- 
try illegally  without  paying  the  required  duty. 

Smuggling  is  a  crime,  for  the  prosecution  and  punishment  of  which  stringent 
laws  are  enacted. 

888.  A  customs  broker  is  a  person  familiar  with  customs  law 
and  practice,  who  makes  entries  and  transacts  similar  business  for 
importers.     He  frequently  acts  as  agent  or  attorney  for  his  principal. 


§§  889-893]  CUSTOMHOUSE   BUSINESS  337 

889.  Tare  is  an  allowance  made  for  the  box,  bag,  crate,  or  other 
covering  of  the  goods.     Leakage  is  the  allowance  made  for  waste 
of  liquids  imported  in  barrels  or  casks. 

890.  The  gross  weight  is  the  weight  before  any  allowances  for 
tare,  etc.,  are  made. 

891.  Net  weight  is  the  weight  after  all  allowances  have   been 
made. 

The  ton  by  law  consists  of  2240  avoirdupois  pounds  in  all  cases  where  it  is 
used  in  the  customs. 

892.  To  find  a  specific  duty. 

893.  Example.     Find  the  specific  duty  on  160  gallons  of  wine  at 
$  2  per  gallon ;  leakage,  10  %  • 

SOLUTION 

10%  of  160  gallons  =  16  gallons  leakage. 

160  gallons  —  16  gallons  =  144  gallons,  the  net  quantity. 

$2  x  144  =  $288,  the  specific  duty. 

WRITTEN  EXERCISE 

1.  Find  the  total  specific  duty  on  1250  bushels  barley  at  30  $  per 
bushel;  400  bushels  onions  at  40^  per  bushel;  1260  pounds  cheese 
at  6^  per  pound  ;  2500  bushels  wheat  at  25^  per  bushel. 

2.  Find  the  total  duty  on  the  following :  900  pounds  unground 
cayenne  pepper  at  2^  per  pound;  1200  bushels  malt   at  45  /  per 
bushel ;  800  pounds  butter  at  6  $  per  pound. 

8.  If  the  duty  on  plate  glass  is  8^  per  square  foot,  how  much 
will  be  the  charge  on  an  importation  of  175  boxes,  each  containing 
25  plates  16  by  24  inches  in  size  ? 

4-  Find  the  specific  duty  on  1656  pounds  macaroni,  at  1 J  t  per 
pound;  900  pounds  hops  at  12^  per  pound;  3150  pounds  filler 
tobacco,  unstemmed,  at  35^  per  pound;  and  165  pounds  hemp 
cordage  at  2  f  per  pound. 

5.    After  being  allowed  10  %  for  leakage,  a  wine  merchant  paid 
$864  duty  at  f  2  per  gallon,  on  12  casks  of  wine.     How  many  gal- 
lons did  each  cask  originally  contain  ? 
MOORE'S  COM.  AR. — 22 


338  PERCENTAGE   AND   ITS  APPLICATIONS        [§§  894-895 

894.  To  find  an  ad  valorem  duty. 

895.  Example.     What  is  the  duty  on  an  invoice  of  leather  goods 
from  Vienna,  the  dutiable  value  being  15,240  crowns  and  the  rate  of 

duty  35%  ad  valorem? 

SOLUTION 

$.203  =  the  value  on  1  crown  in  United  States  money. 

$.203  x  15,240  =  $3093.72,  the  dutiable  value  in  United  States  money. 

35  %  of  $  3094  =  $  1082.90,  the  ad  valorem  duty. 

WRITTEN  EXERCISE 

1.  Find  an  ad  valorem  duty  of  35%  on  an  importation  invoiced 
at  17,450  francs. 

2.  What  is  the  duty  at  50%   ad  valorem,  on  a  consignment  of 
650  dozen  cotton  gloves  invoiced  at  90  francs  per  dozen  ? 

3.  Find  the  duty  at  60%  ad  valorem  on  3  cases  of  silk  goods 
from  Berlin,  invoiced  at  4692  marks  each. 

4.  I  imported  from  England  20  cases  woolen  goods,  weighing 
390  pounds  each;  tare  10%  ;  invoiced  at  £410  per  case.    What  was 
the  total  duty  at  44^  per  pound  and  60%  ad  valorem  ? 

5.  I  received  by  steamer  Raglan  from  Liverpool  the  following 
invoice  of  goods:  768  yards  velvet,  invoiced  at  £1,  12s.  per  yard; 
2150  yards  lace,  invoiced  at  3s.  4d.  per  yard ;  1200  yards  broadcloth, 
invoiced  at  15s.  per  yard ;  3520  yards  carpet,  invoiced  at  11s.  6d. 
per  yard.     If  the  duty  on  the  velvet  was  60%,  on  the  lace   and 
broadcloth  35%,  and  on  the  carpet  50%,  how  much  was  the  total 
duty  to  be  paid  ? 

WRITTEN  REVIEW 

1.  What  is  the  duty  on  1000  yards  Brussels  carpet  27  inches 
wide,  invoiced  at  6s.  9d  per  yard ;  duty  28  ^  per  square  yard  spe- 
cific and  40%  ad  valorem? 

2.  If  the  duty  on  flannel  is  22^  per  pound  specific  and  30%  ad 
valorem,  how   much  must  be  paid   on  an  invoice  of  2150  yards, 
weighing  420  pounds,  and  valued  in  Canada,  whence  it  was  imported, 
at  75  ^  per  yard  ? 

3.  Find  the  duty  at  40%  ad  valorem  on  3  dozen  clocks,  invoiced 
at  $  21.50  each,  and  6  dozen  watches,  invoiced  at  $  35  each. 


895] 


CUSTOMHOUSE   BUSINESS 


339 


4-   Find  35%  ad  valorem  duty  on   250  cases  German  toys  in- 
voiced at  175  marks  per  case. 

5.  I  imported  from  Belgium  300  meters  Brussels  carpet,  f  of  a 
yard  wide,  at  5  francs  per  meter.     I  paid  a  specific  duty  of  28  ^  per 
square  yard  and  an  ad  valorem  duty  of  30%.     What  was  the  total 
duty? 

NOTE.     A  meter  is  equivalent  to  1.0936  yards. 

6.  What   is   the   amount   of   duty  chargeable   on  4000  pounds 
worsted  yarn  invoiced  at  £490,  when  the  rate  of  duty  is  38^^  per 
pound  and  40%  ad  valorem  ? 

7.  A  merchant  imported  10  gross  table  knives  costing  15s.  per 
dozen   in   Sheffield,  England.     What  was   the   duty  at  $2.40  per 
dozen  specific  and  40%  ad  valorem  ? 

Find  the  dutiable  value  and  compute  the  duty  on  the  following 
entries  of  merchandise : 

8 


Manifest  No.. 


INWARD  FOREIGN  ENTRY  OF  MERCHANDISE 


Imported  by; 


tf      / 
rT^^^ 


.Master, 


:  In  the  steamer 
Arrived  _^ 


Number* 


Packages  and  Contents 


Quantity 


Duty 


Total 


t  **-** 


**,** 


340  PERCENTAGE   AND   ITS   APPLICATIONS        [§§  895-896 

9. 


Manifest  No.. 


Invoiced 


at  yW 


INWARD  FOREIGN  ENTRY  OF  MERCHANDISE 

-^   /         ^/-JP      *~X          T  ^y' 

Imported  by  /J^f^Sz^Zstf^f  yl^-^r^f?^ In 


.     Fr 


Arrived 


Packages  and  Contents 


Quantity 


Duty 


Total 


10.  A  merchant  imports  1200  yards  Brussels  carpet,  |  of  a  yard 
wide,  invoiced  at  £200.    Compute  a  duty  of  28  ^  per  square  yard  and 
an  ad  valorem  duty  of  40%.     If  freight  charges  and  losses  aggre- 
gated $  185.50,  at  what  price  per  yard  must  the  carpet  be  sold  to 
gain  20%? 

11.  A  Boston  merchant  imported  mandolins  invoiced  in  Germany 
at  40  marks  each.     If  he  paid  an  ad  valorem  duty  of  45%,  what 
price  must  he  sell  them  for  to  gain  20%  on  the  cost? 

EXCHANGE 

896.  Exchange  treats  of  methods  of  making  payments  at  distant 
places  without  the  transmission  of  money. 

Settlements  are  effected  by  means  of  written  orders  called  bills  of  exchange, 
express  money  orders,  telegraphic  money  orders,  letters  of  credit,  etc.  and  the 
risk  and  expense  of  sending  the  money  itself  is  avoided. 


§§  897-901]  EXCHANGE  341 

897.   An  exchange  center  is  some  recognized  money  center 

The  principal  exchange  centers  of  the  United  States  are  New  York,  Boston, 
Philadelphia,  Chicago,  St.  Louis,  Baltimore,  Cincinnati,  and  San  Francisco  ; 
of  Europe,  London,  Paris,  Antwerp,  Geneva,  Amsterdam,  Hamburg,  Frankfort, 
Berlin,  and  Vienna. 

.   Exchange  is  of  two  kinds :  domestic,  or  inland,  and  foreign. 


DOMESTIC  EXCHANGE 

899.  Domestic  exchange  is  exchange  payable  in  the  country  in 
which  it  is  drawn. 

Domestic  bills  of  exchange  are  commonly  called  drafts. 
The  business  of  making  payments  by  means  of  drafts  and  bills  of  exchange 
is  usually  conducted  through  the  medium  of  banks  and  bankers. 

900.  Funds  may  be  remitted  from  one  place  to  another  place  in 
the  same  country  in  six  different  ways  without  the  transmission  of 
money,  as  follows : 

1.  By  a  postal  money  order.  4.  By  a  bank  draft. 

2.  By  an  express  money  order.  5.  By  a  check. 

3.  By  a  telegraphic  money  order.     6.  By  a  sight  draft  of  a  cred- 

itor on  a  debtor. 

901.  A  postal  money  order  is  an  order  drawn  by  the  postmaster, 
or  his  clerk,  at  one  office,  directing  the  postmaster  of  another  office 
to  pay  to  the  person  named  in  his  private  letter  of  advice  the  sum 
specified  in  the  order. 

Applications  for  postal  money  orders  must  be  in  writing,  and  must  state  the 
amount  of  each  order  wanted,  the  name  and  address  of  the  person  to  whom 
the  order  is  to  be  paid,  and  the  name  and  address  of  the  remitter. 

At  the  present  time  the  maximum  amount  for  which  a  single  postal  money 
order  may  be  issued  is  $  100,  and  the  rates  charged  are  as  follows  : 

$2.50  or  less  ....     3^.  $30.00  to  $40.00    .  .  .  15?. 

$2.50  to  $5.00     .     .     .     5?.  $40. 00  to  $50. 00     .  .  .  18?. 

$ 5.00  to  $10. 00  .     .     .     8?.  $50.00  to  $60. 00     .  .  .20?. 

$10. 00  to  $20. 00      .     .  10?.  $60. 00  to  $75.00    .  .  .  25?. 

$20.00  to  $30.00      .     .  12?.  $75. 00  to  $100. 00  .  .  .  30?. 

The  payee  who  desires  a  money  order  to  be  paid  to  another  person  must  fill 
out  and  sign  the  form  of  transfer  which  appears  on  the  face  of  the  order.  More 
than  one  transfer  is  prohibited  by  law. 


342 


PERCENTAGE   AND   ITS  APPLICATIONS         [§§  901-904 


If  a  money  order  is  lost,  a  certificate  should  be  obtained  from  both  the 
paying  and  issuing  postmasters  stating  that  it  has  not  been  paid  and  will  not  be 
paid.  The  Post  Office  Department  at  Washington  will  then  issue  another  order 
upon  application. 

902.  An  express  money  order  is  an  order  drawn  by  the  agent  of 
the  express  company  at  any  given  office  directing  another  agent  of 
the  company  at  some  designated  place  to  pay  to  the  person  named 
therein  a  certain  sum  of  money. 

Express  money  orders  may  be  obtained  for  any  number  of  dollars,  and  the 
rates  at  the  present  time  are  the  same  as  for  postal  money  orders. 

Express  money  orders  are  transferable  by  indorsement,  the  same  as  notes, 
checks,  etc. 

903.  A  telegraphic  money  order  is  an  order  drawn  by  a  telegraph 
agent  at  any  given  office  instructing  the  agent  at  some  designated 
office  to  pay  to  the  person  named  in  the  telegraphic  message  the  sum 
specified,  upon  his  personal  application  and  proper  identification. 

At  the  present  time  telegraphic  transfer  rates  are  as  follows  : 

$60  or  less      .     .     .     60^.  $200  to  $300    .     .     $1.50. 

$50  to  $100   .     .     .     1%.  $300  to  $400    .     .     $1.75. 

$100  to  $200.     .     .     $1.25.  $400  to  $500    .     .     $2.00. 

Over  $  500,  special  rates. 

The  rates  in  the  above  table  are  entirely  apart  from  the  cost  of  telegraphic 
service,  which  is  based  upon  distance  and  the  number  of  words  contained  in  the 


904.  A  bank  draft  is  an  order  written  by  one  bank  directing 
another  bank  to  pay  a  specified  sum  of  money  to  a  third  party,  or  to 
his  order. 


§§  905-909]  EXCHANGE  343 

905.  Nearly  all  banks  keep  money  deposited  with  some  other 
bank,  called  a  correspondent,  at  one  or   more   commercial   centers 
against  which  they  draw  drafts  to  sell  to  their  customers  for  remit- 
tance purposes.     These  drafts  pass  as  cash  in  the  section  tributary 
to  the  commercial  centers  upon  which  they  are  drawn. 

Banks  usually  make  a  charge  called  exchange  for  the  trouble  ol  keeping 
funds  on  deposit  at  commercial  centers  and  drawing  drafts  against  these  funds. 
These  charges  range  from  rL%  to  £%.  On  many  small  drafts  a  definite  charge 
ranging  from  10  $  to  50  ?  is.  frequently  made.  Some  banks  make  no  charge  for 
drafts  sold  to  regular  depositors. 

906.  Instead  of  making  remittances  by  bank  drafts  merchants 
frequently  send  their  personal  checks  in  payment  of  bills. 

907.  A  check  is  an  order  on  a  bank  by  a  depositor  for  the  pay- 
ment of  money ;  except  that  it  is  drawn  by  a  person,  it  is  very  much 
like  a  bank  draft. 


Rochester,  N.  V-,  ^f^>^i/> ^    Vt*.t.   No.. 

ALLIANCE  NATIONAL  BANK 

Pay  to  the  order  of. 

Dollars. 


908.  Commercial  drafts  play  a  prominent  part  in  facilitating  the 
payment  of  bills  at  distant  places. 

Commercial  drafts,  which  include  sight  and  time  drafts,  were  discussed  on 
pages  260  and  261. 

Exercises  in  discounting  time  drafts  were  given  in  connection  with  bank 
discount,  pages  269  and  270. 

909.  Formerly  domestic  exchange  was  at  a  premium  or  discount 
in  the  city  where  purchased  according  as  the  balance  of  trade  between 
that  city  and  the  one  on  which  the  draft  was  drawn  was  in  favor  of 
or  against  the  former  city.     If  the  drawer  city  owed  the  drawee  city, 
exchange  on  the  latter  would  be  at  a  premium  at  the  former  place; 
if  the  balance  of  trade  was  in  favor  of  the  drawer  city,  the  co  .dition 


344  PERCENTAGE   AND   ITS  APPLICATIONS        [§§909-912 

of  exchange  would  be  reversed  in  the  two  places.    For  a  number  of 
years  past,  however,  domestic  exchange  has  been  practically  at  par. 

910.  Bankers  usually  make  a  charge  called  collection  for  collect- 
ing out-of-town  drafts  deposited  with  them. 

911.  Sometimes  unaccepted  time  drafts  are  left  with  a  bank  for 
collection,  and  sometimes  they  are  offered  for  discount. 

912.  Banks  are  usually  willing  to  accept  for  discount  the  unac- 
cepted drafts  of  responsible  parties  when  they  are  properly  indorsed. 

WRITTEN  EXERCISE 

1.  W.  J.  Boone  &  Co.,  of  San  Francisco,  Cal.,  have  bills  to  pay  as 
follows:  T.  W.  Brooke,  Dayton,  O.,  $650;  E.  L.  Grey  son  &  Sun, 
Cedar  Eapids,  la.,  $  46.53;  Barnes  &  Snyder,  Bolton,  Mo.,  $48.50; 
and  their  traveling  salesman,  W.  H.  Post,  is  wanting  $100  for  ex- 
penses at  Denver,  Col.     They  pay  the  amounts  by  remitting  as  fol- 
lows :  T.  W.  Brooke  and  E.  L.  Grey  son  &  Son,  express  money  orders; 
Barnes  &  Snyder,  postal  money  order;    and  W.  H.  Post,  by  tele- 
graphic transfer  in  a  ten-word  message.     If  the  cost  of  the  telegram 
was  50^,  what  was  the  total  amount  required? 

2.  Barnum  &  Co.,  of  St.  Paul,  drew  a  sight  draft  of  $1400  on 
Martin  &  Cole,  415  High  St.,  Boston,  on  account  of  an  invoice  of 
hides  shipped  to  them  valued  at  $3000,  as  per  bill  of  lading  attached 
to  the  draft.    They  sold  the  draft  at  the  bank  at  |%  discount.    What 
were  the  proceeds  ? 

8.  A  commission  merchant  of  Charleston,  S.C.,  bought  a  ninety- 
day  commercial  draft  at  1%  discount  for  $800  drawn  on  a  Boston 
firm.  If  money  be  worth  6%,  what  did  the  draft  cost  him  ? 

SOLUTION 

$  .015  =  the  bank  discount  on  $  1  for  90  da. 

$.005  =  the  commercial  discount  on  $  1. 

$  .015  +  $  .005  =  $  .02,  the  total  discount  on  $  1. 

$  1  -  $  .02  =  $  .98,  the  proceeds  of  $  1. 

$  .98  x  800  =  $  784,  the  cost  of  the  draft. 


§§  912-913]  EXCHANGE  345 

4.  A  commission  merchant  holds,  subject  to  the  order  of  his 
principal,  §5005.     His  principal  directs  him  to  remit  the  amount  by 
New  York  draft  after  deducting  the  cost  of  the  draft.     If  the  bank 
charges  exchange  at  the  rate  of  ^%,  what  will  be  the  face  of  the 
draft  ? 

5.  Gates  &  Son,  of  Memphis,  drew  a  sight  draft  on  Perrin  & 
Boon,  Portland,  Me.,  for  $8750.85,  which  they  sold  at  the  Cotton 
Exchange  Bank  at  f  %  discount.     How  much  were  the  proceeds  ? 

6.  I  drew  a  60-day  draft  on  one  of  my  customers  and  sold  it 
to  a  broker  at  f  %  discount,  receiving  $  1354.18  as  proceeds.     What 
was  the  face  of  the  draft,  money  being  worth  6%  ? 

7.  Jno.  W.  Williams,  of  Boston,  Mass.,  remitted  Janis  Bros. 
&  Co.,  of  Milwaukee,   $1750  by  draft   on  New  York,  exchange 
15^  per  each  $100;  Martin  &  Co.,  of  Allentown,  Pa.,  by  American 
Express  money  order,  $  89.75 ;   and  Theodore  Emens,  $  28.50,  by 
post  office  money  order.     Find  the  total  cost  of  exchange. 

8.  Thomas,  Bailey  &  Co.,  of  St.  Louis,  drew  a  sight  draft  for 
$  1900  on  Slocum,  Wilde  &  Co.,  291  Milk  St.,  Boston,  Mass.,  on  ac- 
count of  an  invoice  of  molasses  shipped  them,  valued  at  $3506,  as 
per  bill  of  lading  attached  to  draft.     They  sold  the  draft  at  a  bank 
at  \°/o  discount.     What  were  the  proceeds  ? 

9.  A  wholesale  grocer  owed  for  an  invoice  of  $5425.40,  pur- 
chased in  New  York,  subject  to  a  discount  of  6%  if  paid  within  10 
days.     Within  the  required  time  he  discounted  the  bill  and  remitted 
for  balance  as  follows :   A  sight  draft   which  he  bought  of  E.  M. 
Brooks  on  Gunn  &  Baker  for  $4000,  at  \°/0  discount,  and  a  bank 
draft  for  the  remainder,  the  exchange   being  10^  for  each  $100. 
How  much  was  required  to  settle  the  bill,  and  how  much  was  gained 
by  discounting  it  ? 

10.  Hedman  &  Son  drew  a  60-day  draft  on  Johnson  Manufactur- 
ing Co.  for  $2500,  and  had  it  discounted  at  a  bank  at  6%.  If  the 
rate  of  collection  was  J  %,  what  were  the  proceeds  of  the  draft  ? 

FOREIGN  EXCHANGE 

913.  Foreign  exchange  is  exchange  payable  in  another  country 
than  that  in  which  it  is  drawn.  It  is  by  means  of  the  system  of 
foreign  exchange  that  the  people  of  the  various  nations  pay  their 
debts  to  one  another. 


346  PERCENTAGE  AND  ITS   APPLICATIONS          [§§  914-917 

914.  The  business  of  foreign  exchange  was  brought  about  by  the 
fact  that  goods  are  exported  and  imported  by  the  nations  of   the 
earth,  and  the  fact  that  investors  put  money  in  the  enterprises  and 
securities  of  nations  foreign  to  their  own. 

During  the  year  1906  the  goods  imported  by  the  United  States  amounted  to 
$1,226,563,843,  and  the  goods  exported  to  $1,717,953,382;  hence  that  year 
foreign  countries  owed  us  $491,389,539  for  goods  we  exported  in  excess  of  what 
we  owed  them  for  goods  imported.  The  imports  and  exports  were  paid  for 
chiefly  through  the  medium  of  foreign  exchange.  This  same  means  is  used  in 
paying  the  sums  invested  by  Americans  in  foreign  countries,  the  sums  invested 
by  foreigners  in  this  country,  the  amounts  spent  by  Americans  abroad  and 
by  foreigners  traveling  in  this  country,  and  the  sums  of  money  sent  by  foreigners 
in  this  country  to  their  families  in  Europe.  The  grand  total  of  our  foreign 
exchange  is  thus  extremely  large. 

915.  The  following  are  some  of  the   more   common   forms   of 
foreign  exchange:   a  draft,  check,  bill  of  exchange,   money   order, 
circular  letter  of  credit,  traveler's  cheque,  an  order  (either  written  or 
cabled)   to  pay  certain   persons   money   in   some  foreign   country. 
The.  transportation  of  gold  or  specie  from  one  country  to  another  is 
also  an  important  part  of  the  system  of  foreign  exchange. 

Exchange  of  any  kind  whatever  may  be  made  payable  in  the  money  of  the 
country  in  which  payment  is  to  be  received,  or  in  the  money  of  a  country  in  which 
a  great  financial  city  is  located.  London  is  the  greatest  financial  center  of  the 
world,  and  many  drafts  on  Germany,  France,  Norway,  Sweden,  Russia,  China, 
India,  and  other  countries  are  payable  in  sterling  exchange.  New  York  is 
the  financial  center  of  all  America,  and  many  drafts  or  bills  of  exchange  drawn 
on  Canada,  Mexico,  and  the  countries  of  South  America  are  payable  in 
New  York  funds. 

916.  A  bill  of  exchange  is  a  draft  drawn  payable  in  a  foreign 
country.   If  it  is  drawn  to  cover  the  value  of  goods  exported,  a  bill  of 
lading  and  an  insurance  certificate  usually  accompany  it ;  such  a  bill 
is  known  as  a  documentary  bill  of  exchange.     If  the  bill  of  lading  and 
the  insurance  certificate  are  not  attached,  the  bill   of   exchange  is 
known  as  a  clean  bill  of  exchange. 

917.  Frequently  bills  of  exchange  are  drawn  in  sets  of  two  or 
sometimes  three,  called  first,  second,  and  third  of  exchange.     When 
bills  are  drawn  in  sets  of  two  they  are  sent  by  different  mails,  so 
that  if  one  is  lost  the  other  may  be  presented.     If   three  bills  are 
drawn,  the  third  one  is  kept  by  the  purchaser  as  a  memorandum. 


§§  918-020]  EXCHANGE  34? 

A  SET  OF  EXCHANGE 


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YV/Sf//S/{'~7>-is/.j/  /////////// J/ssf//  /<"  ///c  X////V  /'// 


918.  Bills  of  exchange  are  many  times  used  as  a  means  of  col- 
lecting debts  due  in  foreign  countries.     The  method   employed  is 
similar  to  that  used  in  collecting  debts  by  means  of  commercial 
drafts. 

919.  The  par  of  exchange  is  the  established  value  of  the  stand- 
ard unit  of  money  of  one  country  expressed  in  that  of  another.     It 
is  of  two  kinds :  intrinsic  and  commercial. 

920.  The  intrinsic  par  of  exchange  is  the  real  or  intrinsic  value 
of  the  coins  of  one  country  as  compared  with  those  of  another. 

Thus,  the  pound  sterling  of  Great  Britain  contains  113  grains  of  pure  gold, 
and  the  dollar  of  the  United  States  contains  23.22  grains  of  pure  gold.  Since 
113  grains  are  4.8665  times  23.22  grains,  the  pound  sterling  is  worth  $4.86^. 


348  PERCENTAGE   AND  ITS  APPLICATIONS         [§§921-927 

921.  The  commercial  par  of  exchange,  commonly  called  the  course 
of  exchange,  is  the  market  value  of  the  standard  unit  of  money  of 
one  country  in  the  currency  of  another. 

922.  The  course  of  exchange  is  usually  governed  by  the  relative 
state  of  indebtedness  of  merchants  of  different  countries  and'  the  sup- 
ply of  gold  and  silver  ;  hence  it  may  be  at  a  premium  or  at  a  discount. 

If  the  merchants  of  England  owe  the  merchants  of  the  United  States  more 
than  is  usual,  exchange  in  London  on  New  York  quickly  advances  to  a  high  rate, 
while  exchange  in  New  York  on  London  declines. 

923.  Normally  foreign  exchange  rates  fluctuate  between  gold- 
exporting  and  gold-importing  points.     If  it  costs  less  to  export  gold 
than  to  buy  a  bill  of  exchange,  the  gold  itself  is  shipped.     However, 
gold  is  seldom  actually  shipped  in  quantities  of  less  than  $  20,000. 

If  the  rate  paid  for  demand  sterling  in  New  York  drops  below  $4.84|  per 
pound,  it  is  cheaper  to  import  gold  in  large  quantities  than  to  have  remittances 
made  from  abroad  by  means  of  draft.  If  exchange  in  New  York  goes  above 
$4.88£  per  pound  sterling,  then  gold  can  be  exported  at  less  cost  than  to  buy  a 
draft  on  London. 

924.  In  the  exportation  and  importation  of  gold  there  must  be 
considered  the  risk  of  loss  by  shipwreck,  the  loss  because  of  abrasion, 
the  cost  of  transportation,  and  the  charge  for  insurance. 

The  cost  of  sending  gold  from  New  York  to  London  is  usually  as  follows  : 
Insurance  and  freight,  each  £%;  abrasion  on  $5  gold  pieces,  £%;  on  $10  gold 
pieces,  £%;  on  $20  gold  pieces,  &%  to  $  %. 


925.  Quotations  of  foreign  exchange  are  given  by  means  of  market 
value  equivalents,  no  reference  being  made  to  the  intrinsic  par  value. 

926.  Exchange  on  Great  Britain  is  quoted  by  giving  the  exchange 
value  of  £  1  in  dollars  and  cents. 

Thus,  when  exchange  on  London  is  quoted  at  4.86|,  a  bill  for  £1  will  cost 
•4.86|. 

927.  Exchange  on  France,  Belgium,  and  Switzerland  is  usually 
quoted  by  giving  the  exchange  value  of  $  1  in  francs. 

Thus,  when  exchange  on  Paris  is  quoted  at  5.18,  $1  will  buy  5.18  francs. 
Notice  that  such  quotations  differ  from  all  others  in  character  ;  a  change  from 
to  5.19  is  Si  fall  in  the  rate. 


§§  928-933]  EXCHANGE  349 

928.  Exchange  on  the  Netherlands  is  quoted  by  giving  the  ex- 
change value  of  1  guilder,  or  florin,  in  cents. 

When  exchange  on  Amsterdam  is  quoted  at  41,  1  guilder  is  equal  to  41  cents. 

929.  Exchange  on  Germany  is  quoted  by  giving  the  exchange 
value  of  4  reichsmarks  in  cents. 

When  exchange  on  Berlin  is  quoted  at  96£,  4  reichsmarks  are  equal  to  96^  cents. 

930.  Dealers  in  foreign  exchange  refer  to  changes  in  the  quota- 
tions as  being  of  so  many  points.     A  point  is  one  hundredth  of  a 
cent,  or  one  unit  in  the  fourth  decimal  place. 

If  the  quotation  for  London  exchange  is  4.8255,  the  addition  of  75  points 
would  make  it  4.833. 

931.  Quotations  are  sometimes  made  in  the  form  93^-  +  ^  01 
93J  —  3*2-,  or  5.17  —  y1^,  or  the  like.     The  fraction  means  the  fraction 
of  1  %  of  the  rate.     A  -}-  fraction  always  makes  the  rate  higher  ; 
a  —  fraction,  lower. 

Thus  93|  -  &  means  $  .935  -  ^  %  of  $  .935,  or  $  .93471  ;  but  5.17  -  ^  means 
Fr.  5.17  +  ^%  of  Fr.  5.17,  or  Fr.  5.17323.    See  927. 


932.  The  law  of  England  requires  that  revenue  stamps  be  affixed 
to  all  drafts  drawn  for  more  than  five  days'  sight.  Revenue 
stamps  are  required  also  on  drafts  for  more  than  three  days'  sight 
on  Holland,  and  on  drafts  at  one  or  more  days'  sight  on  France  or 
Germany;  but  the  three  days'  sight  "letters  of  delegation"  on 
Germany  are  exempt.  The  required  amount  of  stamps  is  -^  %  of 
the  face  of  the  draft  ;  their  cost  may  therefore  be  included  by  adding 
to  the  quoted  rate. 


983.  A  cable  transfer  is  a  telegraphic  order  to  pay  a  certain  per- 
son in  a  foreign  country  a  certain  sum  of  money.  Such  transfers 
are  made  by  cipher  codes.  The  rate  charged  for  a  cable  transfer  is 
the  quoted  or  market  rate  of  exchange,  plus  a  commission  of  about 
i%,  plus  the  telegraph  and  cable  charges. 

By  cable  transfer,  a  merchant  who  desires  to  ship  wheat  to  London  can  com- 
plete the  transaction  in  a  few  hours.  He  can  ship  the  wheat,  telegraph  the  fact  to 
the  consignee  atLondon,  obtain  particulars  concerning  the  condition  of  the  market, 
and,  if  he  thinks  best,  have  the  wheat  sold  at  once,  "  to  arrive,"  and  the  proceeds 
remitted  through  a  London  banker.  A  bill  does  not  appear  in  the  transaction. 

A  very  large  amount  of  business  in  foreign  exchange  is  done  by  means  of 
the  cable.  Operators  in  this  line  conduct  their  'business  upon  a  very  close  mar- 


350  PERCENTAGE  AND   ITS   APPLICATIONS          [§§933-936 

gin,  and  calculate  the  outcome  of  their  transactions  to  a  nicety.  This  is  possible 
because  there  is  little  delay  during  which  the  rates  might  materially  change  so 
as  to  cause  a  loss.  During  a  part  of  each  day  the  cable  enables  persons  in  New 
York,  Philadelphia,  Chicago,  London,  Berlin,  and  other  money  centers,  to  con- 
clude transactions  without  material  delay. 

934.  Arbitrage   of   exchange   is   the   calculation   of  the  relative 
values  of  exchange  at  the  same  time  at  two  or  more  places  with  the 
purpose  of  taking  advantage  of  the  difference  in  price.     It  is  con- 
ducted (largely  and  most  profitably  by  cable)  by  buying  simulta- 
neously in  the  cheaper  and  selling  in  the  dearer  market. 

Suppose  that  a  merchant  owes  a  debt  of  £  1500  in  London,  and  that  direct 
exchange  on  London  is  4.87£,  and  direct  exchange  on  Paris  is  Fr.  5.24|  to  the 
dollar,  and  that  Paris  exchange  on  London  is  Fr.  25  to  the  pound.  Then  Fr.  37,500 
in  Paris  will  purchase  £1500.  Fr.  37,500  will  cost,  in  U.S.  money  at  5.24$, 
$7149.67.  Direct  exchange  on  London  at  $4.87|  for  £1500  =  $7312.50; 
7312.50  —  $7149.67  =  $  162.83,  amount  gained  by  arbitrage  or  indirect  exchange. 

If  the  German  mark  could,  because  of  some  condition,  purchase  more  ster- 
ling in  proportion  in  Berlin  than  could  the  franc  in  Paris,  then  the  merchant 
would  gain  still  more  by  buying  marks  at  the  lowest  rates.  But  if  the  mark 
could  be  had  for  less  by  remitting  via  Paris  than  by  direct  exchange  on  Berlin, 
the  payment  of  the  £  1500  would  be  made  in  London  through  both  Paris  and 
Berlin. 

935.  A  letter  of  credit  is  a  circular  letter  issued  by  a  banking 
house  to  a  person  who  desires  to  travel  abroad.     The  letter  is  usu- 
ally addressed  to  the  foreign  correspondents  of  the  bank  issuing  it, 
requesting  them  to  furnish  the  traveler  such  funds  as  he  may  require 
up  to  the  aggregate  amount  named  in  the  letter. 

When  the  traveler  desires  funds,  he  goes  to  any  correspondent  mentioned 
in  the  letter  of  credit,  and  draws  a  draft  on  that  correspondent  for  the  amount 
desired.  The  draft  is  signed  in  the  presence  of  the  correspondent,  who  care- 
fully compares  the  signature  with  the  one  on  the  letter,  and,  if  they  are  found 
to  agree,  the  draft  is  cashed  and  the  amount  inscribed  on  the  back  of  the 
letter.  The  last  draft  drawn  is  attached  to  the  letter  itself. 

The  difference  between  a  bill  of  exchange  and  a  letter  of  credit  is  that  the 
former  is  payable  at  a  certain  designated  place,  at  a  specified  time,  and  in  one 
amount,  while  the  latter  is  payable  at  several  places,  at  different  times,  and  in 
variable  amounts. 

936.  Travelers'  cheques  are  a  substitute  for  letters  of  credit  and 
bills  of  exchange.     They  are  similar  in  form  to  bank  bills.     They 
are  issued  for  fixed  printed  amounts,  with  the  equivalent  of  each 


§§  936-940]  EXCHANGE  851 

denomination  in  the  money  of  the  principal  European  countries,  and 
are  payable  to  order,  after  being  signed  and  countersigned  by  the 
purchaser  or  holder.  They  are  cashed  without  discount  or  com- 
mission by  an  extended  list  of  banks  and  bankers,  and  are  received 
in  settlement  of  hotel  bills  by  the  principal  hotels  in  Europe. 


000000 


Travelers'  Cheque 

937.  To  find  the  cost  of  a  foreign  bill  of  exchange. 

938.  Examples.      1.   Find  the  cost  of  a  draft  on  London   for 
£  380  10s.  6d.  sterling,  exchange  being  quoted  at  $  4.86f  . 

SOLUTION 

£380  10s.  6d.  =  £  380.525. 

4.86|  =  the  market  quotation  of  £  1. 

-#4.86|  X  380.525  =  $  1851.73,  the  cost  of  the  draft. 

2.   Find  the  cost  of  a  sight  draft  on  Paris  for  3108  francs,  exchange 

being  quoted  at  5.18. 

SOLUTION 
$1  =  5.18  francs. 
3108  francs  -=-5.18  francs  =  600. 
Therefore  a  bill  for  3108  francs  will  cost  $600. 

939.  To  find  the  face  of  a  foreign  bill  of  exchange. 

940.  Example.     The  cost  of  a  bill  of  exchange  on  London  was 
$3654.47.     When  exchange  was  quoted  at  $4.86J,  what  was  the  face 
of  the  bill  ? 


352  PERCENTAGE   AND  ITS  APPLICATIONS  £§940 

SOLUTION 

$4.86|  =  the  market  value  of  £  1. 

$3654.47^$  4.86  }  =751.175. 

Therefore  $3654.47  =  £751.175. 

£.175  =  3s.  6d. 

Hence  the  face  of  the  bill  was  £  751  3s.  6d. 


WRITTEN  EXERCISE 

1.  Find  the  cost  of  a  sight  draft  on  London  for  £400  when 
exchange  is  quoted  at  4.865. 

2.  I  bought  a  bill  of  exchange  on  Paris  and  paid  $  2156.    What 
was  the  face  of  the  bill,  exchange  being  quoted  at  5.17f  ? 

3.  A  New  York  importer  who  owed  a  Dresden  manufacturer 
21,320  reichsmarks,  bought  a  bill  of  exchange  on  Berlin  at  95J,  and 
paid  for  the  same  by  check.     What  was  the  face  of  the  check  ? 

4-   What  is  the  face  of  a  bill  of  exchange  on  London  which  can 
be  bought  for  15807.25  if  quoted  at  $4.85,  brokerage  \%  ? 

5.  An  exporter  sold  through  a  broker  a  bill  of  exchange  on 
Hamburg  at  95f ,  and  received  $  5953.49  as  net  proceeds.     What  was 
the  face  of  the  bill,  brokerage  \%1 

6.  Hibbard  &  Co.,  of  Brooklyn,  purchased  a  bill  of  exchange  on 
London  at  3  days'  sight  for  £  342  12s.  6d.  at  4.86  J.     How  much  did 
the  bill  cost  ? 

s 

7.  An  importer  purchased  a  sixty-day  bill  of  exchange  on  Bre- 
men at  95|  for  $446.20.     What  was  the  face  of  the  bill  ? 

8.  A  New  York  diamond  merchant  purchased  a  bill  of  exchange 
on  Amsterdam  at  3  days'  sight  for  63,892  guilders  at  40£.     What  did 
the  bill  cost? 

9.  I  purchased  a  bill  of  exchange  on  Paris  for  33,250  francs  and 
paid  $6412.72.     What  was  the  course  of  exchange  ? 

10.  An  importer  purchased  a  bill  of  exchange  on  Amsterdam  for 
3575  guilders  and  paid  $1443.41  for  it.  What  was  the  course  of 
exchange  ? 


§§940-944]  EXCHANGE  353 

11.  A  Manchester,  England,  manufacturer  drew  a  bill  of  exchange 
at  3  days'  sight  for  £450  10s.  Scl  on  a  Rochester,  N.Y.,  merchant. 
The  draft  was  presented  to  the  drawee  by  a  local  bank,  and  paid  by 
check.     What  was  the  face  of  the  check,  exchange  being  4.85  J? 

12.  Langdon  &  Perry,  of  New  York,  owed  on  foreign  invoices  as 
follows:  T.  C.  Shepherd  Sons,  London,  £  1800  8s.  ;  J.  L.  Von  Buesche, 
Berlin,  1600  marks  ;  Perrie,  Buzzell  &  Co.,  Paris,  4016  francs  ;  F. 
Gonzalez,  Mexico,  816  dollars.     They  bought  at  their  bank  :  exchange 
on  London  at  4.86£;    on  Berlin,  96^;    on  Paris,  5.19  *-;   on  Mexico, 
79£,  and  issued  one  check  to  cover  the  total  purchase.     What  was 
the  amount  of  the  check  ? 

941.  To  find  the  cost  of  exporting  gold. 

942.  Example.    Strawbridge  &  Clothier,  Philadelphia,  are  quoted 
4.89.L  on  London  for  a  sight  bill  of  £6540.     They  export  the  gold 
instead  of  buying  the  exchange,  paying  1  %  insurance,  1  %  freight, 
T3g  °/0  abrasion.     How  much  did  they  save  by  shipping  the  gold  ? 

SOLUTION 

$4.89|  x  654°  =  $32,013.30,  the  cost  of  the  draft. 
4.8665  =  intrinsic  par  of  exchange. 


4.  8665  =.02129. 
4.8665  +  .02129  =4.88779,  the  rate  for  exporting  gold. 
$4.88779  x  6540  =  $31,966.16,  the  cost  of  gold  shipment. 
$32,013.30  -  $31,966.15  =  $47.15,  saved  by  gold  shipment. 

943.  To  find  the  proceeds  of  a  bill  of  exchange. 

944.  Examples.    1.   John  J.  Rose  &  Co.  sell  to  Brown  Bros.  &  Co. 
a  sight  draft  for  Mk.  1852^%.     What  did  they  receive  at  93T\  ? 

SOLUTION 

$  .93^  =  $.933125  =  the  rate  for  4  marks. 
$.933125  x  1852^  -=-  4  =  $432.22,  the  proceeds. 

2.   J.  Griffith  &  Co.  sold  to  Brown  Bros.  &  Co.  at  5.22}  —  -^  a 
sight  draft  for  Fr.  14,645^.     What  are  the  proceeds  ? 

SOLUTION 

$1  =  Fr.  5.22  J-  -  &  =  Fr.  5.22451.    See  931. 

Fr.  14,645^  -  Fr.  5.22451  =  2803.23.  Therefore  the  proceeds  are  $2803.23 
NOTE.     The  following  examples  show  how  a  banker  determines  what  rate  to 
quote  to  a  customer. 

MOORE'S  COM.  AR.  —  23 


354  PERCENTAGE  AND  ITS  APPLICATIONS  [§944 

S.  On  Jan.  3,  1907,  there  is  drawn  by  the  Baldwin  Locomo- 
tive Works,  Philadelphia,  on  the  London  &  Brazilian  Bank, 
London,  a  commercial  bill  of  exchange  at  90  days  after  sight 
for  £617  8s.  3d.  At  4.865  market  quotation  what  should  a 
banker  pay  for  it,  allowing  for  interest  at  3%,  revenue  stamps, 
and  a  commission  of 


NOTE.      Interest  on  London  exchange  is  reckoned  on  a  basis  of  $4.85  per 
pound  365  days  to  the  year.     English  law  allows  3  days  grace. 

SOLUTION 
Market  rate  on  London  =  $4.865 

Disct.  rate  93  ds.  (3  ds.  grace)  =  4.85  x  .03  x  -fa  =     .03707 
Com  .  £  %  per  pound  =  .02433 

Cost  per  pound  revenue  stamps  ^  %  of  rate  =  .00243       .06383 

The  banker's  quotation  $  4.  80  1  17 

£617  8s.  3d  =  £617.4125. 
$4.80117  x  617.4125  =  $2964.30,  the  sum  the  banker  would  pay. 

4.  A  bill  of  exchange  on  Paris  at  30  days  after  sight  is  drawn  in 
favor  of  Lazard  Freres  for  Fr.  16,764^°^  with  documents.     Interest 
rate,  3%;  market   quotation,  Fr.  5.165;  commission,  \%  of  rate. 

What  are  the  proceeds  ? 

SOLUTION 

Market  quotation  on  Paris  —  Fr.  5.165 

Discount,  30  ds.  at  3  %  =  \  %  of  rate  as  .  01291 

Com.  |%  of  rate  =  .00646 

Revenue  stamps  ^%=  .00258 

The  banker's  quotation  $  1  =  Fr.  5.18695 

Fr.  16,764.70  -$-  Fr.  5.  18695  =  3232.09.     Hence  the  proceeds  are  $  3232.  09. 

5.  A  documentary  bill  of  exchange  is  drawn  on  Amsterdam  at 
60  days  after  sight.     What  rate  per  guilder  can  Brown  Bros.  &  Co. 
offer  for  it,  allowing  for  discount  rate,  3J%;    market  quotation, 

;  commission,  \%',  revenue  stamps,  -fa%  ? 

SOLUTION 

Market  quotation  (40|  +  A)  =  $.401376 

Revenue  stamps  ^%  of  rate  =  .0002 

Discount,  60  ds.  at  3^%  of  rate  =  .00234 

Commission  \  %  of  rate  .0005  _  .00304 

The  banker's  quotation  "  $.398336 


§  944]  EXCHANGE  355 

WRITTEN  EXERCISE 

2.  The  Baldwin  Locomotive  Works  of  Philadelphia  sell  a  sight 
draft  for  £416  9s.  4d.  on  the  London  &  Brazilian  Bank  to  Brown 
Bros.  &  Co.  at  4.81-|-.  What  amount  of  money  does  it  bring  ? 

2.  J.  S.  Griffith  &  Co.  sell  to  Brown  Bros.  &  Co.  at  5.20f  —  ^  a 
draft  on  Credit  Lyonnais,  Havre,  for  Fr.  19,745^.  What  amount 
do  they  receive  for  it  ? 

8.  The  Baldwin  Locomotive  Works  draw  a  draft  on  the  London 
&  Brazilian  Bank  at  90  days  sight  for  £314  3s.  4d.  How  much  will 
Brown  Bros.  &  Co.  pay  for  it ;  market  quotation,  4.826 ;  commission, 
|  % ;  interest  rate,  3  % ;  revenue  stamps,  -£$  %  ? 

4.  John  J.  Eose  &  Co.  sell  to  Brown  Bros.  &  Co.  the  draft  shown 
on  page  347.     Market  rate,  91^;  interest  rate,  3^-%;  commission, 
\°/o',    revenue   stamps,  ¥V%-      What  amount  of  money  do  they 
receive  ? 

NOTE.  Interest  on  German  exchange  is  computed  at  95  cents  per  4  marks, 
360  ds.  to  a  year. 

5.  John  Wanamaker  sells  in  New  York  a  documentary  bill  of 
exchange  for  £640  12s.  6d,  at  90  days  sight,  on  a  London  bank. 
What  should  he  receive,  allowing  for  3  %  interest,  revenue  stamps, 
and  a  commission  ^%,  if  the  quotation  is  $4.835? 

6.  The  Rogers  Locomotive  Works  of  Paterson,  N.J.,  sell  to 
Brown  Bros.  &  Co.  a  documentary  bill  of  exchange  on  a  Paris  bank 
for  Fr.  18,765^%,  drawn  90  days  after  sight.     What  are  the  pro- 
ceeds, if  market  rate  is  5.17J  +  -fa ;  interest  rate,  3  %  ;  commission, 
\  %  ;  revenue  stamp,  -fa%? 

7.  The  Wm.  H.  Hostman  Co.  bought  from  John  Heckemann, 
Bremen,  a  bill  of  goods  amounting  to  $4695.45  on  60  days  time, 
2%%,  10  days.    They  send  him  a  draft  to-day  at  sight,  so  as  to  take 
advantage  of  the  discount.    What  is  the  face  of  the  draft,  in  marks, 
if  demand  exchange  is  94J,  which  includes  broker's  commission  ? 

8.  James  McCreery  &  Co.  sell  a  documentary  bill  of  exchange 
on   a  Geneva  banker  for  Fr    14,764^   at  90   days   after   sight, 
to  Lazard  Freres.     What  are   the  proceeds,  the   rate  of  exchange 
being  5.16^-;  rate  of  interest, 3%  ;  commission,  \°/0  ;  revenue  stamps, 

;  and  cost  of  collecting  and  sending  funds  to  Paris, 


356  PERCENTAGE  AND  ITS  APPLICATIONS  [§  944 

9.  Tiffany  &  Co.  import  an  invoice  of  statuettes  and  bronzes 
from  Florence,  Italy,  amounting  to  14,725^-  lire.  They  buy  a 
three  days  after  sight  draft  of  Brown  Bros.  &  Co.  on  a  Flor- 
ence banker  to  pay  the  bill.  What  will  they  pay  Brown  Bros.  & 
Co.  for  it  if  the  market  rate  is  5.17|-;  commission,  \%\  revenue 
stamps,  -fa  %  ;  discount  rate, 


10.  B.   Altman  &  Co.,  New  York,  are  quoted  4.89f,  which  in- 
cludes commission,  for  a  sight  bill  on  London  for  £7560.     They 
buy  the  gold  and  ship  it,  instead  of  buying  the  draft.     How  much 
is  saved  if  they  pay  |  %  insurance,  \  %  freight,  and  ^  %  abrasion  ? 

11.  You  are  a  clerk  in  the  office  of  Brown  Bros.  &  Co.,  and  a 
merchant  hands  you  a  bill  on  London  for  £  312  3s.  4d  at  60  days 
after  sight;  a  bill  at  90  days  after  sight  on    Havre,  France,  for 
Fr.   15,612^;   a  bill   on   Hamburg  at  30  days   after   sight,   for 
Mk.  3412^;  a  sight  bill  on  Amsterdam  for  1345^  guilders.    The 
posted  rates  are  4.831  on  London  ;  5.16  —  -^  on  France  ;  92  \  on  Ham- 
burg; 40^  +  -^  on  Amsterdam.     The  foreign  rate  of  interest  is  3  %. 
You  are  to  allow  for  a  commission  of  \%>  and  for  revenue  stamps 
on  time  paper,  fa%.     How  much  should  you  pay  the  merchant? 

12.  Altman  &  Co.,  New  York,  are  quoted  a  rate  of  4.89J,  which 
includes   commission,  on   London,   for  a   sight  bill   amounting  to 
£846010s.     Instead  of  buying  the  bill  they  export  the  gold.     If 
they  pay  \%  insurance,  \%  freight,  -fa%  for  abrasion,  how  much 
do  they  gain  or  lose  by  shipping  the  gold  instead  of  buying  the 
exchange  ? 

18.  New  York  pates  on  Amsterdam  are  41-J;  on  Paris  are  5.15J. 
A  merchant  owes  a  debt  of  Fr.  13,450  in  Paris.  He  pays  it  by  re- 
mitting via  Amsterdam  at  Fr.  2.14£  to  the  guilder.  What  is  saved 
by  the  indirect  exchange  (934)  ? 

14-  Mr.  W.  of  Philadelphia  goes  to  London  for  a  visit  and  directs 
his  Philadelphia  broker  to  remit  him  $  10,000.  Brown  Bros.  &  Co. 
quote  London  exchange  at  4.89  and  Berlin  exchange  at  951,  both 
rates  including  the  commission.  In  Berlin,  exchange  on  London  is 
quoted  at  Mk.  20.3  to  the  pound.  How  much  more  or  less  in  English 
money  does  Mr.  W.  receive  by  indirect  exchange  than  by  direct  ex- 
change, if  there  is  a  charge  of  \%  for  remitting  from  Berlin  to 
London  ? 


SHARING 

PROPORTIONAL  PARTS 

945.   Sharing  is  the  process  of  dividing  a  number  into  shares 
proportional  to  other  given  numbers. 

DRILL  EXERCISE 

1.  A  and  B  agree  to  perform  a  certain  piece  of  work  for  $  160. 
If  B  can  earn  $  10  while  A  earns  $  6,  how  much  should  each  receive 

as  his  share  of  the  $160  ? 

SOLUTION 

$160  is  to  be  divided  iiito  shares  proportional  to  6  and  10. 

The  numbers  6  and  10  may  be  regarded  as  shares. 

The  whole  will  then  be  represented  by  6  + 10,  or  16  shares. 

16  shares  =  $  160. 

1  share  =  %  10. 

6  shares  =  $60,  or  A's  part. 

10  shares  =  $  100,  or  B's  part. 

2.  Divide  240  into  shares  proportional  to  8  and  16. 

8.   Divide  $  600  into  shares  proportional  to  1,  3,  and  6. 

4.  Divide  a  profit  of  $  1200  among  three  partners  in  a  business 
in  which  A  invests  $  2  as  often  as  B  invests  $4,  and  C$6. 

6.   Divide  $  150  into  shares  proportional  to  ^  and  £. 

SOLUTION 

Fractions  must  be  similar  to  be  compared. 
\  and  \  =  ft  and  ft,  respectively. 

ft  and  ft  stand  in  the  same  relation  to  each  other  as  8  and  2,  respectively. 
Hence  the  whole  may  be  represented  by  6  shares. 
6  shares  =  $150. 

1  share  =  $30. 

3  shares  =  $90,  or  the  first  part. 

2  shares  =  $60,  or  the  second  part. 

6.   Divide  $  780  among  three  persons,  whose  shares  are  to  be  in 
proportion  to  ^,  £,  and  £. 

857 


358  SHARING  [§  945 

7.  Three  men  engage  in  business.    A  puts  in  $  2000 ;  B,  $3000 ; 
and  (J,  $  4000.     They  gain  $  900.     What  should  be  each  man's  share 
of  the  gain  ? 

8.  A  bankrupt  owes  $500  to  A,  $2000  to  B,  and  $1500  to  C. 
If  his  net  resources  are  $2000,  what  will  each  of  his  creditors 
receive  ? 

9.  Divide  $2100  among  A,  B,  and  C  so  that  A's  part  will  be 
twice  B's  part  and  one  half  of  C's  part. 

10.  Three  boys  bought  a  watermelon  for  24^,  of  which  price 
Charles  paid  9^,  John  8^,  and  Walter  If.  Ralph  offered  24^  for 
one  fourth  of  the  melon,  which  offer  was  accepted  and  the  melon 
divided.  How  should  the  24^  received  from  Ralph  be  divided 
among  the  other  three  boys? 

WRITTEN  EXERCISE 

1.  A  will  provided  that  an  estate  be  divided  among  three  per- 
sons in  proportion  to  their  ages,  which  are  15,  18,  and  20  years, 
respectively.     If  the  amount  received  by  the  youngest  person  was 
$3000,  what  was  the  value  of  the  estate  ?        /o  ^ 

2.  A,  B,  and  C  engage  in  trade  for  one  year.     A  puts  in  $  6000, 
B  $3000,  and  C  $2000.     If  their  gain  for  the  year  is  $4400,  what 
is  each  man's  share  ?  ji^0  0      i  •  -     - 

8.  A,  B,  and  C  rent  a  pasture  for  $  740.  A  put  in  3  cows  for 
4  months ;  B,  5  for  6  months ;  and  C,  8  for  4  months.  How  much 
should  each  pay?  !  ^  £00  ^3.0 

4-  An  estate  was  so  divided  between  A  and  B  that  A's  part 
was  to  B's  part  as  \  is  to  \.  If  B  received  $  2400  more  than  A, 
what  was  the  value  of  the  estate  ?  )  2_  o  0  0 

5.  A  man  bequeathed  his  property  to  his  wife  and  two  daugh- 
ters.    The  wife  received  $5  for  every  $3   received   by  the  elder 
daughter  and  for  every   $2  received   by  the  younger  daughter. 
What  was  the  value  of  the  estate,  the  younger  daughter  having 
received  $1500  less  than  her  sister? 

6.  Divide  the  simple  interest  on  $  2600  for  6  years  3  months  at 
4%  among  A,  B,  and  C  in  proportion  to  ^,  ^,  and  ^,  respectively. 


§§945-951]  PARTNERSHIP  359 

7.  A  man  left  his  property  to  be  divided  among  his  three  sons 
and  two  daughters  in  proportion  to  their  ages.     The  sons  are  aged 
20,  16,  and  10  years,  respectively ;   the  daughters  18  and  8  years, 
respectively.     If  the  share  of  the  youngest  child  was  $  7200,  what 
was  the  value  of  the  property  ? 

8.  So  divide  $30,000  among  A,  B,  C,  and  D,  that  their  portions 
shall  be  to  each  other  as  1^,  3,  4^,  and  6. 

9.  Coe,  Hall,  Tell,  and  Lee  have  a  contract  to  dig  a  ditch  which 
Coe  can  dig  in  35  days,  Hall  in  45  days,  Tell  in  50  days,  and  Lee  in 
60  days.     How  long  will  it  take  all  together  to  do  the  work  ?     If 
f  100  be  paid  for  the  work  and  all  join  till  it  is  completed,  how  much 
should  each  get  ? 

PARTNERSHIP 

946.  Partnership  is  an  association  resulting  from  an  agreement 
between  two  or  more  persons  to  place  their  capital  or  services,  or 
both,  in  some  enterprise  or  business,  and  to  share  the  gains  and  bear 
the  losses  in  certain  proportions. 

947.  Partnerships    may   be    formed   by:     1.    Oral    agreement. 
2.   Written  agreement,  (a)  under  seal,  or  (b)  not  under  seal.    3.  Im- 
plication. 

All  important  partnership  agreements  should  be  in  writing,  and  all  of  the 
conditions  relating  to  the  partnership  should  be  definitely  stated. 

948.  The  partners  are  the  persons  associated  in  any  business. 
Collectively  they  are  called  &jirm,  a  house,  or  a  company. 

949.  Partners  are  of  four  classes :   1.  Real  or  ostensible.     2.  Dor- 
mant, silent^  or  concealed.     3.  Limited.    4.  Nominal. 

950.  A  real  or  ostensible  partner  is  one  who  appears  to  the  public 
to  be,  and  who  actually  is,  a  partner. 

951.  A  dormant  or  concealed  partner  is  practically  a  real  partner 
whose  connection  with  the  partnership  is  concealed  from  the  public. 

A  concealed  partner  is  responsible  for  the  debts  of  a  firm  if  his  connection 
with  the  partnership  becomes  known. 


360  SHARING  [§§  952-902 

952.  A  limited  partner  is  one  whose  responsibility  is  limited 
instead  of  general. 

In  case  of  failure  of  the  firm,  the  general  partner  is  individually  responsible 
for  all  the  debts  of  a  firm,  while  the  limited  partner  is  responsible  only  for  the 
amount  named  in  the  partnership  agreement. 

Limited  partnerships  are  forbidden  by  the  laws  of  some  states.  In  the  states 
where  they  are  permitted,  it  is  generally  provided  that  at  least  one  member  of  a 
firm  must  be  a  general  partner. 

953.  A  nominal  partner  is  one  whose  name  appears  to  the  public 
as  a  partner,  but  who  has  no  investment  and  receives  no  share  of  the 
gains. 

A  nominal  partner  is  responsible  to  third  parties  for  the  debts  of  a  firm. 

954.  The   capital   or   stock  is   the   money  or   other  equivalent 
property  invested  in  the  business. 

Capital  is  frequently  real  estate,  personal  property,  time,  skill,  etc. 

955.  Resources  or  assets  are  the  entire  property  of  a  firm,  includ- 
ing accounts  receivable,  bills  receivable,  etc. 

956.  Liabilities  are  the  entire  debts  of  a  firm. 

957.  The  net  capital  is  the  excess  of  the  resources  over  the 
liabilities. 

958.  The  net  insolvency  is  the  excess  of  the  liabilities  over  the 
resources. 

When  the  resources  of  a  firm  exceed  the  liabilities,  the  business  is  said  to 
be  solvent ;  when  the  liabilities  exceed  the  resources,  the  business  is  said  to  be 
insolvent,  or  bankrupt. 

959.  The  net  investment  of  a  person  is  his  investment  minus  all 
withdrawals  for  personal  use. 

960.  A  business  statement  contains  an  itemized  list  of  all  resources 
and  liabilities,  of  losses  and  gains,  the  present  worth  of  the  business, 
and  the  net  gain  or  net  loss  for  any  given  period. 

961.  The  net  gain  is  the  excess  of  total  gains  over  the  total  losses 
for  any  given  period. 

962.  The  net  loss  is  the  excess  of  total  losses  over  the  total  gains 
for  any  given  period. 


§§  962-964]  PARTNERSHIP  361 

DRILL  EXERCISE 

1.  A  commenced  business  with  a  cash  investment  of  $6000. 
At  the  end  of  the  year  his  net  capital  is  $3500.     What  is  his  net 
gain  or  loss,  no  withdrawals  having  been  made? 

2.  B  began  business  with  an  investment  of  $7500.     At  the  end 
of  one  year  his  net  capital  is  found  to  be  $  9500.     Find  the  net  gain 
or  loss. 

8.  C  began  business  Jan.  1  with  a  cash  investment  of  $2500. 
At  the  close  of  the  year  his  net  insolvency  is  $1700.  Required  the 
net  gain  or  loss. 

4.  At  the  beginning  of  a  year  D's  net  insolvency  was  $600;  at 
the  close  of  the  year  his  net  capital  was  $150.     Required  his  net 
gain  or  loss  for  the  year. 

5.  E's  insolvency  at  the  beginning  of  the  year  was  $  4000,  and 
at  the  close  of  the  year  is  $  3500.     What  has  been  his  net  gain  or 
loss  for  the  year  ? 

6.  F's  loss  for  one  year  is  $  700.     His  insolvency  at  the  end  of 
the  year  is  $  300.     Required  the  net  capital  at  the  beginning. 

7.  G's  net  gain  for  one  year  is  $  2000 ;  his  net  capital  at  the  end 
of  the  year  is  $  1500.     What  was  the  net  capital  or  net  insolvency  at 
the  beginning  of  the  year  ? 

8.  Jan.  1,  1903,  H's  resources  were  $3000,  and  his  liabilities 
$2000;  one  year  later  his  resources  were  $2000  and  his  liabilities 
$  3000.     What  was  the  net  gain  or  loss  for  the  year  ? 

9.  Fs  insolvency  at  the  end  of  one  year  is  $  3000.     If  he  gained 
$  1200  during  the  year,  what  was  his  net  capital  or  net  insolvency  at 
the  beginning  of  the  year  ? 

10.  J's  capital  at  the  end  of  one  year  was  $5200.  If  his  gain 
for  the  year  was  $6900,  what  was  his  net  capital  or  net  insolvency 
at  the  beginning  of  the  year  ? 

963.  To  divide  the  gain  or  loss  when  each  partner's  investment 
has  been  employed  for  the  same  period  of  time. 

964.  Example.     A  and  B  enter  into  partnership  to  carry  on  a 
commission  business  for  one  year,  A  investing  $7000  and  B  $  4000. 


362  SHARING  [§  964 

During  the  year  they  gain  $  3300.   What  should  be  each  man's  share 

of  the  gain  ? 

SOLUTION 

The  total  investment  =  $  7000  +  $  4000,  or  $  11,000. 
A's  investment  =  rVV'oV  or  A  o*  tlie  total  investment. 
B's  investment  =  fWo^  or  T4i  of  tne  total  investment. 

Each  partner  receives  such  a  part  of  the  gain  as  his  investment  is  a  part  of 
the  total  investment.     Therefore, 

A's  share  of  the  gain  =  ^T  of  $  3300,  or  $2100. 
B's  share  of  the  gain  =  £  of  $  3300,  or  $  1200. 


ORAL  EXERCISE 

Find  each  man's  gain  or  loss  in  each  of  the  following  problems, 
the  gains  being  shared  or  losses  borne  in  proportion  to  investments. 

1.  A  invested  $300  and  B  $200;  they  gained  $150. 

2.  C  invested  $  800  and  D  $  300 ;  they  gained  $330. 
S.   E  invested  $1000  and  F  $  800;  they  lost  $  360. 

4.  G  invested  $  1200  and  H  $  900 ;  they  gained  $  560. 

5.  I  invested  $  900  and  J  $  700;  they  lost  $  320. 

6.  K  invested  $1500  and  L  $  1200;  they  gained  $540. 

7.  M  invested  $2000  and  N  $  800;  they  lost  $560. 

8.  0  invested  $1000  and  P  $500;  they  gained  $  186. 

WRITTEN   EXERCISE 

1.  A  and  B  unite  in  the  purchase  of  a  house  costing  $  4200 ;  A 
pays   $1800   and   B   $2400.      The   property   rents   for   $294   per 
annum.     What  share  of  the  rent  ought  each  to  receive  ? 

2.  A,  B,  and  C  enter  into  partnership  for  the  purpose  of  carry- 
-^r)  ing  on  a  manufacturing  business.     A  joint  capital  of  $65,000  is 

formed  of  which  A  furnishes  f,  B  |  of  the  remainder,  and  C  what 
still  remains.  Their  net  gain  for  one  year  is  equivalent  to  25  % 
of  the  net  capital  invested.  Required  each  man's  share  of  the 
net  gain. 


§  964]  PARTNERSHIP  3G3 

S.  Two  men  bought  a  mine  for  $  20,000,  of  which  sum  A  paid 
112,500  and  B  the  remainder.  They  sold  the  mine  for  $42,000. 
How  much  of  the  gain  should  each  man  receive  ?  What  part  of  the 
selling  price  should  each  receive  ? 

4.  A,  B,  and  C  engage  in  business,  A  investing  $  1250 ;  B,  $  750 ; 
and  C,  $1000.     They  gain  $  957.30.     How  much  of  the  gain  should 
each  receive  ?     How  much  is  each  man  worth  after  receiving  his 
share  of  the  gain  ? 

5.  B,  C,  and  D  unite  their  capital  amounting  to  $12,000  in  a 
business  venture  and  realize  a  gain  of  $1500,  of  which  B  received    '- 
$750;    C,   $500;    and   D,   $250.     What  was   the   investment  of ^ 
each? 

6.  In  a  partnership  A  invested   $5000  and  received  f  of  the 
gain ;  B  invested  $  2400  and  received  -^  of  the  gain*     The  gains  and 
losses  were  as  follows:  merchandise,  gain,  $840.30;  expense,  loss, 
$  310.40 ;  real  estate,  gain,  $  265.61.    What  was  the  net  gain  ?    What 
was  the  gain  of  each  partner  ?     After  receiving  his  share  of  the  gain, 
what  was  each  partner  worth  ? 

7.  In   a  partnership   C   invested   $4000  and  D  $2000.     It  is 
agreed  that  if  gains  are  realized  they  are  to  be  shared  according  to 
the  investment,  each  partner  receiving  such  part  of  the  gain  as  his 
investment  is  a  part  of  the  total  investment;    that  if  losses  occur 
they  are  to  be  borne  equally.     At  the  end  of  one  year  the  gains 
and  losses  were  as  follows:   merchandise,  gain,  $534.20;  expense, 
loss,  $325.60;   real  estate,  loss,  $675;   interest,  gain,  $34.25;  dis- 
count, loss,  $  56.35.     What  was  the  net  gain  or  loss  ?     What  was 
each  partner  worth  after  his  share  of  the  net  results  of  the  business 
was  carried  to  his  account  ? 

8.  In  a  partnership  E  invested  $7654,  F  $8000,  and  G  $7000; 
the  gains  and  losses  were  to  be  shared  equally.     During  a  month  the 
gains  and  losses  were  as  follows  :  merchandise,  gain,  $  2318 ;  stocks 
and  bonds,  gain,  $735;  expense,  loss,  $1140;  interest,  gain,  $342. 
What   was   the   net   gain?     What  was  each  partner's  interest  at 
closing  ? 

9.  H's  net  loss  for  one  year  was  $17,290.     His  insolvency  at 
the  close  of  the  year  was  $10,000.     Find  the  net  capital  at  the 
beginning. 


364 


SHARING 


[§§  965-967 


965.  To  divide  the  gain  or  loss  according  to  the  amount  invested 
and  the  time  the  investment  is  employed. 

966.  The  best  method  of  solving  partnership  problems  is  to  treat 
them  from  the  accountant's  standpoint.     In  connection  with  the  so- 
lution of  each  problem  a  ledger  page  may  be  used,  an  account  opened 
with  each  partner,  the  net  gain  or  loss  properly  divided,  and  the 
ledger  accounts  with  the  partners  closed. 

967.  Examples.     1.   In  a  partnership,  A  invested  $2000  for  8 
months,  B,  $3000  for  6  months,  and  C,  $4000  for  5  months.     A 
gain  of  $  1350  was  realized.     Find  the  gain  of  each  partner. 


SOLUTION 

Dr.                                                            A                                                             Cr. 

Present  worth 

2400 

2000 

Net  Gain 

400 

= 

2400 

2400 

Dr. 


B 


Cr. 


Present  worth 

3450 

Net  Gain 

3000 
450 

3450 

3450 

Dr. 


Cr. 


_ 

Present  worth 

4500 

Net  Gain 

4000 
500 

4500 

4500 

A's  investment,  $  2000  for  8  mo.  =  $  16,000  for  1  mo. 

B's  investment,  $  3000  for  6  mo.  =     18,000  for  1  mo. 

C's  investment,  $4000  for  5  mo.  =     20,000  for  1  mo. 

Firm's  investment  =  $  54,000  for  1  mo. 

Each  partner  should  receive  such  apart  of  the  gain  as  his  investment  for  1  mo. 
is  a  part  of  the  total  investment  for  1  mo.    Therefore, 

A's  gain  =  JV  OflHfR)  of  •  1350,  or  $  400. 
B's  gain  =  &  or  J  G^)  of  $  1350,  or  $450. 
C's  gain  =  tf  (§$${{$)  of  $  1350,  or  $  500. 


967] 


PARTNERSHIP 


365 


2.  A  and  B  were  partners  in  a  manufacturing  business.  Their 
investments  and  withdrawals  for  one  year  were  as  follows :  Jan.  1, 
A  invested  $2000;  Apr.  1  he  withdrew  $500;  July  1  he  invested 
$1000;  Oct.  1  he  withdrew  $700.  Jan.  1,  B  invested  $3000; 
May  1  he  withdrew  $1000;  Sept.  1  he  invested  $500.  During 
the  year  the  firm  gained  $1780.  What  was  each  man's  share  of 


the  gam?                                   SOLUTION 

Dr.                                                            A                                                             Cr. 

Apr. 

1 

500 

Jan. 

1 

2000 

Oct. 

1 

700 

July 

1 

1000 

Jan. 

1 

Present  worth 

£580 

Jan. 

1 

Net  Gain 

780 

3780 

3780 

Dr. 


Cr. 


May 

1 

1000 

Jan. 

1 

3000 

Jan. 

1 

Present  worth 

3500 

Sept. 

1 

500 

Jan. 

1 

Net  Gain 

1000 

4500 

4500 

A's  Investment 

$  2,000  for  3  mo.  =    $  6,000  for  1  ino. 
1,500  for  3  mo.  =       4,500  for  1  mo. 
2,500  for  3  mo.  =       7,500  for  1  mo. 
1,800  for  3  mo.  =       5,400  for  1  mo. 
A's  total  investment  =  $  23,400  for  1  mo. 

B's  Investment 

$3,000  for  4  mo.  =  $  12,000  for  1  mo. 
2,000  for  4  mo.  =       8,000  for  1  mo. 
2,500  for  4  mo.  =     10,000  for  1  mo. 
B's  total  investment  =  $  30,000  for  1  mo. 
$23,400  +  $30,000  =  $53,400,  the  firm's  investment  for  1  mo. 
A's  gain  =  f  f  of  $  1780,  or  $  780. 
B's  gain  =  f$  of  $  1780,  or  $  1000. 

WRITTEN  EXERCISE 

1.   Three   persons   traded   together  and   gained   $900.      A   in- 
vested in  the  business  $  1000  for  6  months ;   B  invested  $  750  for 
10  months ;  and  C  invested  $  1200  for  5  months.     How  should  the  ^ 
gain  be  divided  ? 


2  7 


366  SHAKING  [§§  967-969 


2.   A,  B,  and  C  were  partners.     A  had  $  800  in  the  business  for 
^1  year,  B  had  $1000  in  for  9  months,  and  C  had  $2000  in  for  8 
^  ^  ^months.     How  should  a  gain  of  $2150  be  divided  ? 

f  7   V  /?       A    A,  B,  and  C  hired  a  pasture  for  6  months  for  $  95.10;  A  put 
£  .-     in  75  sheep,  and  2  months  later  took  out  40 ;  B  put  in  60  sheep,  and 


at  the  end  of  3  months  put  in  45  more  ;  C  put  in  200  sheep,  and  after 
4  months  took  them  out.     What  part  of  the  rent  should  each  pay  ? 


4.  A  commenced  digging  a  ditch,  and  after  wording  6  days  was 
joined  by  B,  after  which  the  two  worked  together  9  days,  when  they 

^  2  were  joined  by  C.  The  three  then  worked  12  days,  and  at  the  end 
of  that  time  A  left  the  job  and  D  worked  with  the  other  two  3  days, 

.   r      when  the  work  was  completed.-    If  $  92  was  paid  for  the  work,  how 

much  should  each  receive  ? 

/ 
v5.   Martin  and  Eaton  were  partners  one  year,  Martin  investing 

at  first  $  5000  and  Eaton  $  3000  ;  after  6  months  Martin  drew  out 
$  3000  and  Eaton  invested  $  1500  ;  they  gained  $  3600.  What  was 
the  gain  of  each  and  the  present  worth  of  each,  at  the  time  of  the 
dissolution  of  the  partnership  ? 

'-  6.  A  and  B  engaged  in  a  grocery  business  for  3  years  from 
March  1,  1901,  On  that  date  each  invested  $  1600  ;  June  1,  of  the 
same  year,  A  increased  his  investment  $400,  and  B  withdrew  $300) 
Jan.  1,  1902,  each  withdrew  $1000;  Jan.  1,  1903,  each  invested 
$  1500.  How  should  a  gain  of  $  7500  be  divided  at  the  expiration 
of  the  partnership  contract  ? 


PARTNERSHIP  SETTLEMENTS 

968-  A  partnership  settlement  is  an  adjustment  of  the  net  value 
of  a  business  among  the  partners  when  a  partnership  is  dissolved, 
either  by  mutual  consent  or  by  limitation  of  contract. 

969.  In  most  partnership  affairs  a  business  statement  would  bo 
required  in  connection  with  the  finding  of  the  condition  of  the  busi- 
ness at  the  close  of  any  given  period. 

WRITTEN  EXERCISE 

1.  Copy  the  following  statement  of  resources  and  liabilities,  fill- 
ing out  the  missing  terms : 


§969] 


PARTNERSHIP 


• 
367 


000 


.^/00 


2-000 


t  To  be  written  in  red  ink. 


368  SHARING  [§  069 

Using  the  foregoing  statement  form  as  a  model,  make  statements 
to  solve  the  questions  embodied  in  each  of  the  following  problems  : 

2.  At  the  time  of  closing  business,  the  resources  of  a  firm  were : 
cash,  $931.50;  merchandise,  per  inventory,  $13,196.25;  notes  and 
accounts  due  it,  $8154;    interest  on  same,  $211.50;    real   estate, 
$11,150.      The  firm  owed,  on  its  notes,  acceptances,  and  bills  out- 
standing, $7142,  and  interest  on  the  same,  $348.50;  and  there  was 
an  unpaid  mortgage  on  the  real  estate  of  $  2500,  with  interest  accrued 
thereon  of  $88.50.     If  the  invested  capital  was  $  22,500,  what  was 
the  net  solvency  or  insolvency  of  the  firm  at  closing,  and  how  much 
has  been  the  net  gain  or  net  loss  ? 

3.  Burke,  Brace,  and  Baldwin  became  partners,  each  investing 
$15,000,  and  each  to  have  one  third  of  the  gains  or  sustain  one  third 
of  the   losses.     Burke  withdrew  $2100   during  the  time   of  the 
partnership,  Brace  $  1800,  and  Baldwin  $  2000.     At  the  close  of  busi- 
ness their  resources  were  :  cash,  $  3540 ;  merchandise,  $  14,785 ;  notes, 
acceptances,  and  accounts  receivable,  $  16,250 ;  real  estate,  $  28,500. 
They  owed  on  their  outstanding  notes  $8125,"  and  on  sundry  personal 
accounts  $1950.     Find  the  present  worth  of  each  partner  at  closing.  ( 

4.  Parsons  and  Briggs  became  partners  Apr.  1,  1901,  under  an 
agreement  that  each  should  be  allowed  6%  simple  interest  on  all 
investments,  and  that,  on  final  settlement,  Briggs  should  be  allowed 
10%  of  the  net  gains,  before  other  division,  for  superintending  the 
business,  but  that  otherwise  the  gains  and  losses  be  divided  in  pro- 
portion  to  average  investment.      Apr.  1,  1901,  Parsons   invested 
$18,000,  and  Briggs  $4000;  Jan.  1, 1902,  Parsons  withdrew  $5000, 
and  Briggs  invested  $3000;  Aug.  1,  1902,  Briggs  withdrew  $1500; 
Dec.  1,  1902,  the  partners  agreed  upon  a  dissolution  of  the  partner- 
ship, having  resources  and  liabilities  as  follows : 

Resources  Liabilities 

Cash  on  hand  and  in  bank,       $  1,101.05  Notes  and  acceptances,  $6,520.00 

Accounts  receivable,  16,405.50  Outstanding  accounts,    21,246.50 

Bills  receivable,  2,550.00  Rent  due,  1,200.00 

Interest  accumulated  on  same,        287.41 
Mdse.  per  inventory,  9,716.55 

If  only  80%  of  the  accounts  receivable  prove  collectible,  what  has 
been  the  net  gain  or  net  loss  ?  What  has  been  the  net  gain  or  net 
loss  of  each  partner  ?  What  is  the  firm's  net  insolvency  at  dissolu- 
tion ?  What  is  the  net  insolvency  of  each  partner  ? 


§  909] 


PARTNERSHIP 


369 


Liabilities 

Mortgage  on  real  estate,  $  7,000.00 

Interest  accrued  on  same,  210.00 

Notes  outstanding,  26,950.00 

Interest  accrued  on  same,  811.75 

Due  Barnes,  Clay  &  Co.,  33,560.00 


5.  A  and  B  became  partners  for  one  year,  A  investing  f  of  the 
capital,  and  B  f  ,  the  agreement  being  that  the  gains  or  losses  shall 
be  apportioned  according  to  average  net  investment,  and  that  each 
partner  be  allowed  6%  interest  per  annum  on  all  investments,  and 
be  charged  interest  at  the  rate  of  6%  on  all  sums  withdrawn.     At 
the  end  of  the  year  the  firm  had  resources  and  liabilities  as  follows  : 

Resources 

Mdse.,  per  inventory,  $  21  ,460.00 

Real  estate,  15,000.00 

Cash,  1,950.00 

Bills  receivable,  13,146.50 

Interest  accrued  on  same,  519.25 

Accounts  due  the  business,  11,218.50 

Furniture,  1,320.00 

Delivery  wagons  and  horses,  2,100.00 

It  is  found  that  33  \°fo  of  the  accounts  due  the  firm  are  uncol- 
lectible. If  the  firm's  losses  during  the  year  have  been  $12,000, 
how  much  was  invested  by  each  partner  ?  What  is  the  present 
worth  or  net  insolvency  of  the  firm,  and  of  each  partner,  at  closing  ? 

6.  Mason  and  Kivers  were  joint  partners,  each  investing  an  equal 
part  of  $  9000.     At  the  end  of  the  year  the  resources  and  liabilities 
were  as  follows:  cash  on  hand,  $2212.45;  merchandise  on  hand, 
$7278.54;  bills  receivable,  $943.50;  interest  accrued  on  bills  receiv- 
able, $22.70;  office  safe,  $160;  accounts  receivable,  $2956.20;  5% 
of  the  accounts  receivable  were  estimated  as  not  collectible  ;  accounts 
payable,  $1147;  unpaid  freight  bill,  $64.50;  bills  payable  $560; 
interest  accrued  on  bills  payable,  $17.25.     What  was  the  net  gain  ? 


What  was  each  partner's  present  worth  at  the  close  of  the  year  ? 


7.  D  and  E  are  partners,  each  investing  $9000.  The  losses 
and  gains,  respectively,  are  to  be  borne  or  shared  equally.  At  the 
end  of  one  year  the  following  is  a  list  of  their  resources  and  liabili- 
ties :  merchandise  on  hand,  $  6235.42  ;  personal  accounts  due  the 
firm,  $4785.15;  cash  on  deposit,  $4756.20;  cash  in  safe,  $543.82; 
bills  receivable  on  hand,  $2658.90;  N.  Y.  C.  &  H.  E.  E.  E.  stock, 
$  3600  ;  First  National  Bank  stock,  $  2000  ;  bills  payable  outstand- 
ing, $4298.75;  due  sundry  persons  on  account,  $3215.60.  During 
the  year  D  withdrew  $1800  and  E  withdrew  $  2000.  What  is  the 
net  gain  and  the  present  worth  of  each  partner  ? 


•w  f^i. 
•  v- 


^  SHARING  -  [§9«0 

llr^ 

<£  'A  and  B  are  equal  partners  in  a  business,  the  losses  and  gains 

of  which  are  to  be  borne  and  shared  equally.  The  following  is  the 
condition  of  the  business  at  the  close  of  one  year..  Resources  :  cash, 
$  4275 ;  merchandise,  per  inventory,  $  5476..20 ;  accounts  receivable, 
$  2356 ..75.  Liabilities :  bills  payable,  $  2240  j  Davis  &  Weller,  $  140Q. 
The  net  gain  for  the  year  is  $4267^5.  What  was  the  investment 
of  each  partner  at  the  beginning  of  business  ? 

9.   The  following  is  the  trial  balance  of  the  firm  of  Austin  & 
Leland,  at  the  close  of  one  year's  business : 


Dr. 

Charles  Austin,  Proprietor,  $350.00 

William  Leland,  Proprietor,  327.00 

Cash,  5,647.27 

Merchandise,  3,187.56 

Bills  receivable,  9,000.00 

Bills  payable, 

Expense,  475.00 

Interest,  76.46 

Discount, 

Accts.  receivable,  3,427.50 

Accts.  payable, 


Or. 

$0,885.24 
6,385.24 


5,909.00 

64.65 
3,756.65 


Inventories : 


Mdse.  (goods  on  hand) , 
Mdse.  (unpaid  freight  bill), 

Expense  (coal  on  hand), 
Expense  (unpaid  gas  bill), 


$8,764.50 
82.26 

$28.46 
10.50 


It  is  estimated  that  10%  of  the  accounts  receivable  cannot  be 
collected.  Make  a  business  statement.  What  is  the  present  worth 
of  the  business?  What  is  the  net  gain?  What  is  the  present 
worth  of  each  proprietor? 

rf?<S  10.  E.  H.  Hill  was  associated  in  business  with  N.  P.  Pond  for 
one  year,  each  investing  $5600.  At  the  end  of  the  year  their  books 
show  the  following :  "Resources  :  mdse.,  $  4500  ;  notes,  $  2500  ;  ac- 
counts receivable,  $13,000;  cash,  $1500.  Liabilities:  notes,  $5000; 
accounts  payable,  $1250;  due  Pond  for  special  services,  $400. 
Find  the  net  loss  or  net  gain,  which  divide  equally  between  the 
partners,  and  then  find  each  partner's  present  worth. 


§§970-974]  BUILDING   AND  LOAN   ASSOCIATIONS  371 


BUILDING   AND   LOAN  ASSOCIATIONS 

970.  Building  and  loan  associations  were  organized  originally  for 
the  purpose  of  assisting  members  to  build  homes  with  their  savings. 
The  funds  of  such  an  association  are  loaned,  from  time  to  time,  on 
satisfactory  real  estate  security,  to  that  member  who  will  pay  the 
highest  premium  for  the  loan  in  addition  to  the  regular  interest. 

971.  The  Department  of  Labor  outlines  a  building  and  loan  asso- 
ciation as  follows :    "  The  stockholder  pays  a  stipulated  minimum 
sum,  say  one  dollar,  when  he  takes  his  membership  and  buys  his 
share  of  stock.     He  then  continues  to  pay  a  like  sum  each  month 
until  the  aggregate  sums  paid,  augmented  by  the  profits,  amount  to 
the  maturing  value  of  the  stock,  usually  $200;  and  at  this  time  the 
stockholder  is  entitled  to  the  full  maturing  value  of  the  share  and 
surrenders  the  same. 

"It  is  seen  clearly,  then,  that  the  capital  of  a  building  loan  association  con- 
sists of  the  continued  savings  of  its  members  paid  to  the  association  upon  shares 
of  stock,  increased  by  the  interest  and  premiums  which  the  association  has  re- 
ceived from  loans  made  by  it  from  the  savings  of  its  members  thus  paid  to  the 
association,  and  from  all  other  sources  of  income.  The  amount  of  the  capital 
of  the  association  increases  from  month  to  month. 

972.  "Shares  are  usually  issued  in  series.    When  a  second  series 
is  issued,  the  issues  of  a  previous  series  cease. 

"  The  term  during  which  a  series  is  open  for  subscriptions  is  usually  either 
three  months,  six  months,  or  twelve  months. 

973.  "  Before  a  share  matures  it  has  two  values :  the  book  value  is 
ascertained  by  adding  all  the  dues  that  have  been  paid  to  the  profits 
that  have  accrued ;  that  is,  the  book  value  is  the  actual  value.     The 
withdrawal  value  is  that  amount  which  an  association  is  willing  to 
pay  a  stockholder  who  desires  to  sever  his  connection  with  the  asso- 
ciation prior  to  the  date  upon  which  his  share  matures." 

974.  Every  building  and  loan  association  adopts  its  own  by-laws. 
A  few  by-laws  of  one  association  may  be  summarized  as  follows  : 

(1)  Provision  is  made  for  monthly  dues  of  $  1  and  maturing  value  of  $  200  per 
share,  as  suggested  in  971. 

(2)  Every  stockholder  may  receive  a  loan  of  $200  for  each  share  of  stock  held 
by  him  or  her,  to  be  secured  by  mortgage  on  satisfactory  real  estate ;  but  every  loan 
shall  be  awarded  to  the  highest  bidder  at  stated  meetings,  and  the  bid  shall  be  at 


372 


SHARING 


[§§  974-976 


so  many  cents  per  share,  which  premium ,  together  with  interest  at  the  rate  of  six  per 
cent  per  annum,  shall  be  paid  monthly  during  the  continuance  of  the  series.  (When 
the  series  matures,  the  loan  is  canceled  by  the  maturing  value  of  the  stock.) 

(3)  During  the  first  year  of  any  series  the  withdrawal  value  of  the  stock  thereof 
shall  be  the  amount  paid  in  as  dues,  less  all  fines  and  charges  and  a  tax  of  fifteen 
cents  per  share ;  after  the  first  year  of  any  series  the  withdrawal  value  of  the  stock 
thereof  shall  be  the  amount  paid  in  as  dues  with  such  sum  in  addition,  not  exceeding 
two  thirds  of  the  actual  profits,  as  may  be  ordered  from  time  to  time  by  the  Board 
of  Directors,  less  all  fines  and  charges. 

975.  When  a  member  fails  to  pay  his  dues  on  any  meeting 
night,  a  fine  is  charged,  which  he  must  pay  in  addition  to  the  dues. 

Sometimes  the  fine  is  a  certain  sum  (as  10  cents)  per  share  ;  sometimes  2  % 
per  month  interest  on  the  amounts  unpaid.  Under  the  latter  rule,  for  example, 
if  the  owner  of  10  shares  neglected  to  pay  his  $  10  monthly  dues  for  5  succes- 
sive meetings,  he  would  owe,  at  the  sixth  meeting,  $  60  in  dues  and  also  $  3  in 
fines  ;  the  fines  having  been  charged  against  him  as  follows  ;  $  .20  (2%  of  $  10) 
at  the  second  meeting  ;  $  .40  additional  (2  %  of  $ 20)  at  the  third :  $.60  more  at 
the  fourth  ;  $.80  more  at  the  fifth ;  and  $  1  more  at  the  sixth. 

976.  The  following  shows  a  form  of  roll  book  for  one  month. 
The  other  months  of  the  fiscal  year  are  extended  across  the  page : 


No. 

SHAKES 

BOOK 
NUMBER 

NAMB 

MARCH 

Loans 

Dues 

Interest 
and 
Premiums 

Fines 

Total 

Amount 
Paid 

16 

467 

George  Brown 

$1200 

00 

$15 

00 

$6 

00 
60 

$21 

00 

10 

463 

Amos  Burnett 

20 

00 

$1 

00 

21 

00 

15 

395 

Charles  Cook 

15 

00 

1 

50 

16 

50 

10 

465 

John  Dercum 

2000 

00 

10 

00 

10 

3 

00 
00 

23 

00 

5 

466 

Edward  Enstice 

10 

00 

50 

10 

50 

12 

442 

Frank  French 

12 

00 

12 

00 

In  the  illustration  George  Brown  has  borrowed  $1200,  for  which  he  pays 
interest  at  6%  per  annum,  and  a  premium  of  10  cents  per  share  per  month  on 
the  6  shares  covered  by  the  loan.  This  makes,  with  his  dues,  a  total  of  $21.60 
due  from  him  for  March.  Amos  Burnett  owes  dues  for  February  and  March 
and  a  fine  of  10  cents  per  share  on  unpaid  February  dues.  John  Derciun's  pre- 
mium is  30  cents  a  share. 


§§997-979]  BUILDING  AND  LOAN  ASSOCIATIONS  873 

977.  To  find  the  withdrawal  value. 

978.  Example.     B  owns  10  shares  in  a  first  series  and  has  paid 
dues  for  60  months,  during  which  he  has  been  a  member.     What 
amount  can  he  now  withdraw,  if  he  is  allowed  annual  profits  of  3  %  ? 

SOLUTION 

$  10  x  60  =  $  600,  total  dues  paid  in. 

The  first  $  10  draws  profits  for  60  months,  the  second  $  10  for  69  months, 
and  so  on  ;  hut  the  custom  is  to  allow  an  average  of  one  half  the  total  time,  or 
30  months,  in  this  case,  on  the  total  amount. 

$600  x  .03  x  fg  =  $45,  profit  allowed. 
$600  +  $45  =  withdrawal  value. 

WRITTEN  EXERCISE 

1.  K  has  25  shares  in  a  first  series  and  has  paid  in  dues  for 
8|  years.     If  there  are  no  fines  or  charges  against  him,  how  much 
can  he  withdraw,  if  allowed  6  %  annual  profit  ? 

2.  L  owns  15  shares  in  a  series,  having  been  a  member  9  years. 
He  has  paid  his  dues,  but  owes  $  1.20  for  fines.    How  much  can  he 
withdraw,  if  allowed  4  %  annual  profit  ? 

8.  M  holds  20  shares  in  a  series  and  has  paid  in  his  dues  for  6 
years,  but  there  are  fines  of  $  4.60  charged  against  him.  How  much 
can  he  withdraw,  if  allowed  profits  at  the  rate  of  3  %  per  annum  ? 

979.  Under  the  by-laws  in  974,  the  following  table  shows  how 
the  money  from  dues  will  probably  come  in  and  be  paid  out  for 
twenty-four  months,  in  a  series  of  300  shares,  where  all  dues  are 
paid  promptly. 

The  association  receives,  at  the  first  meeting,  $800  dues ;  one  month  later, 
$300  dues  and  $  1.50  accrued  interest  on  the  first  loan  of  $300,  which  was  made 
at  the  first  meeting ;  at  the  third  meeting,  $  300  dues  and  $  3  interest  for  one 
month  on  total  loans  of  $600,  and  so  on.  At  the  twenty-first  meeting,  the 
interest  received  has  accumulated  to  the  amount  of  $315,  so  that  month  the 
association  makes  a  loan  of  twice  the  usual  amount.  Notice  the  effect  of  this 
extra  loan  in  the  other  columns  for  the  following  month. 

The  profit  per  share  is  determined  by  dividing  the  interest  by  the  number  of 
shares.  The  book  value  of  a  share  at  any  meeting  is  found  by  adding  to  the 
previous  book  value  the  dues  and  profit  for  one  month.  The  table  takes  no 
account  of  premiums  or  fines,  which  in  practice  will  sometimes  greatly  increase 
the  receipts,  profits,  and  book  values. 


374 


SHARING 


£§979 


DUES  REC'I) 
EACH  MONTH 

INTEREST 
KEC'D  EACH 
MONTH 

AMOUNT 

LOANED 

EACH  MONTH 

PROFIT  PER 

SHARE 

BOOK  VALITB 

OF   EACH 

SHARE 

1st  meeting  .... 

300 

300 

1 

00 

2d  meeting    .... 

300 

1 

60 

300 

005 

2 

005 

3d  meeting    .         .    . 

800 

3 

00 

300 

01 

3 

015 

4th  meeting  .... 

300 

4 

50 

300 

016 

4 

03 

6th  meeting  .... 

300 

6 

00 

300 

02 

6 

05 

6th  meeting  .... 

300 

7 

50 

300 

025 

6 

075 

7th  meeting  .         .    . 

300 

9 

00 

300 

03 

7 

105 

8th  meeting  .... 

300 

10 

50 

300 

035 

8 

14 

9th  meeting  .... 

300 

12 

00 

300 

04 

9 

18 

10th  meeting  .... 

300 

13 

50 

300 

045 

10 

226 

llth  meeting  .... 

300 

15 

00 

800 

05 

11 

276 

12th  meeting  .... 

800 

16 

50 

300 

055 

12 

33 

13th  meeting  .... 

300 

18 

00 

300 

06 

13 

39 

14th  meeting       .     .     . 

300 

19 

50 

800 

065 

14 

455 

15th  meeting  .... 

300 

21 

00 

300 

07 

15 

525 

16th  meeting  .... 

300 

22 

50 

800 

075 

16 

60 

17th  meeting            .     . 

800 

24 

00 

300 

08 

17 

68 

18th  meeting  .... 

300 

25 

50 

300 

085 

18 

765 

19th  meeting  .... 

800 

27 

00 

800 

09 

19 

855 

20th  meeting  .     . 

800 

28 

50 

800 

095 

20 

95 

21st  meeting  .... 

800 

30 

00 

600 

10 

22 

05 

22d  meeting    .... 

800 

33 

00 

300 

11 

23 

16 

23d  meeting    .    . 

300 

34 

50 

800 

115 

24 

275 

24th  meeting  .     .     . 

300 

36 

00 

300 

12 

25 

395 

25th  meeting  .... 

300 

37 

50 

800 

125 

26 

52  . 

In  this  table,  the  owner  of  6  shares  pays  $6  a  month  dues.  Suppose  now 
that  he  borrows,  to  the  limit  of  his  stock,  the  sum  of  $  1200  (taking  from  the 
association  $300  a  month  for  four  months),  paying  a  premium  of  25  cents  a 
share,  what  would  he  pay  monthly  thereafter?  In  addition  to  the  $6  dues 
each  month  he  will  have  to  pay  $1.50  premium  and  $6  monthly  interest  on  the 
money  borrowed  —  a  total  of  $  13.50.  These  payments  will  continue  till  his  6 
shares  accumulate  the  full  value,  $  1200,  when  both  the  shares  and  the  loan  are 
canceled. 

WRITTEN  EXERCISE 

1.  Make  out  a  table  like  that  above,  covering  a  period  of  six 
months  (7  meetings),  for  a  new  series  made  up  of  550  shares. 

2.  Make  out  a  table  for  twelve  months  for  a  third  series  of  400 
shares,  reckoning  interest  (including  premiums)  at  6.6%. 


§§979-080]  BUILDING   AND   LOAN  ASSOCIATIONS  375 

8.  Formulate  a  table  for  a  fourth  series  of  800  shares  for 
twenty-four  months,  reckoning  interest  at  9  %. 

4.  B  owns  8  shares  of  the  series  in  the  table  on  page  374.  At 
the  end  of  the  twenty-fourth  month  (i.e.  at  the  twenty-fifth  meeting), 
what  is  the  to'al  profit  on  his  holdings  ?  What  is  their  book  value  ? 

d.  How  much  would  B  have  to  pay  each  month  in  dues,  pre- 
mium, and  interest  if  he  owns  8  shares  and  borrows  to  the  limit 
of  his  stock,  having  bid  30  cents  per , share  premium  ? 

6.  Suppose  Y  bids  20  cents  premium  for  a  loan  to  the  limit  of 
his  10  shares.  What  are  his  monthly  payments  thereafter  if  he 
secures  the  loan? 

980.  To  find  the  profit  per  share  in  several  series.  In  the  table 
on  page  374  but  one  series  is  shown,  and  the  finding  of  profits  is 
relatively  a  simple  matter.  When  a  new  series  is  started  every 
three  months  or  every  six  months  or  every  year,  the  profits  must  be 
distributed  equitably  among  the  stockholders  of  the  different  series. 

For  example,  suppose  that  a  series  was  started  each  year,  and  that  we  are 
now  at  the  end  of  the  fourth  year.  In  the  first  series,  the  first  dollar  paid  in 
on  each  share  has  run  for  48  months ;  the  second,  47  months,  etc.,  to  the  last 
dollar,  which  has  run  but  1  month  ;  it  is  the  custom  to  take  the  average  time, 
therefore,  as  24  months.  In  the  second  series,  the  average  time  will  be  con- 
sidered  18  months ;  in  the  third,  12  months ;  and  in  the  fourth,  6  months. 

Suppose  A  holds  110  shares  and  has  been  a  member  48  months  ;  B  holds  88 
shares  and  has  been  a  member  36  months ;  C  holds  66  shares  and  has  been  a 
member  24  months ;  and  D  holds  300  shares  and  has  been  a  member  12  months  ; 
and  that  at  the  end  of  the  fourth  year  there  is  a  total  profit  of  $  1246.40  to  be 
divided  between  them.  Eaoh  one's  portion  of  the  profit  would  be  adjusted  as 
follows : 

Payments       Shares  Totals  ^n^nSr*  For  one  Month 

A  $48     x     110     =    85280.  $5280     x     24  =        $126720 

B      36     x      88     =       3168.  3168     x     18  =  67024 

C      24     X       66     =        1584.  1584     x     12  =  19008 

D      12     X     800    a       8600.  8600     x      0  =  21600 

Total  invested  one  month     ....    $224352 

Gain 

8hares    per  Share 

A's  part  is  $fflM  of  $  1246.40  «  $704.00.  $704.00  -t- 110  =  $6.40 

B's  part  is  fjfffo  of  $  1246.40  =    316.80.  316.80  -4-    88  =    8.60 

C's  part  is  jffffo  of  $  1246.40  =     105.60.  105.60  +    66  =     1.60 

D's  part  is  ^P&  of  $  1246.40  =     120.00.  120.00  -*-  300  =       .40 

Total  gain      .  $1246.40. 


376  SHARING 

The  preceding  work  shows  clearly  the  principles  involved  in  the  distribution 
of  profits.  In  practice,  however,  the  work  is  simplified  by  using  what  are  known 
as  •* earning  powers"  and  "  lowest  terms,11  as  follows : 

Average      Earning  Lowest  Lrnnat     Total 

Payments    Time  In       Power  Term*  Shares    Terms    Powers 

Months     per  Share  per  Share 

Istser.    $48    x    24  =   $1152.    1152 -*- 72  =  16.            110  x  16=  1760 

2d  ser.       36    x    18  =        648.      648  +  72  =   9.              88  x     9  =  792 

3d  ser.        24    x    12  =        288.      288  •*-  72  =    4.               66  x     4  =  264 

4th  ser.      12    X     6  »          72.        72  +  72  a    1.            800  x     1  =  300 

Grand  total  8116 
Total  gain,  $  1246.40  -<-  8116  =  40  cents,  the  gain  per  power. 

The  gain  per  power  multiplied  by  the  lowest  term  for  a  series  gives  the  gain 
per  share  in  that  series.  That  is : 

$.40  x  16  as  $6.40,  gain  per  share,  1st  series 
$.40  X  9  as  3.60,  gain  per  share,  2d  series 
$  .40  x  4  a  1.60,  gain  per  share,  3d  series 
$  .40  x  1  SB  .40,  gain  per  share,  4th  series 
A's  gain  is  therefore  $6.40  x  110,  etc. 

WRITTEN  EXERCISE 

1.  L  has  owned  120  shares  of  stock  for  36  months;  M  has 
owned  75  shares  for  30  months ;  N  has  owned  90  shares  for  24 
months;  O  has  owned  250  shares  for  12  months;  P  has  owned  150 
shares  for  6  months.     What  will  be  each  one's  share  of  a  gain  of 
$966.35?    What  are  the  lowest  terms  of  the  earning  powers  per 
share  ?    What  is  each  one's  profit  per  share  ? 

2.  Q  owns  275  shares,  R  owns  150  shares,  S  owns  55  shares,  T 
owns  44  shares,  U  owns  33  shares,  and  V  owns  25  shares  in  a  loan 
association.      R  has  paid  dues  for  6  months,  S  24  months,  T  18 
months,  and  Q,  U,  and  V  12  months.     How  shall  they  share  a  gain 
of   $262.01?     What  are  the  lowest  terms  of  the   earning  powers 
per  share  ?     What  is  each  one's  profit  per  share  ? 

S.  K  owns  50  shares,  upon  which  he  has  paid  dues  for  60  months; 
W  owns  75  shares  and  has  paid  dues  for  48  months ;  X  owns  90 
shares  and  has  paid  dues  for  36  months;  Y  owns  200  shares  and 
has  paid  dues  for  24  months ;  Z  owns  40  shares  and  has  paid  dues 
for  12  months.  If  there  is  a  gain  of  $1390.48,  what  part  of  it 
belongs  to  X? 


RATIO   AND   PROPORTION 

RATIO 

981.  The  ratio  of  two  numbers  is  their  relative  greatness  as 
expressed  by  the  quotient  of  the  first  divided  by  the  second. 

Thus,  the  ratio  of  8  to  4,  commonly  written  8  : 4,  is  8  -r-  4,  or  2  ;  the  ratio  of 

7  :  9  is  7  -T-  9,  or  $  ;  etc. 

982.  Since  ratio  is  simply  relative  greatness,  no  ratio  can  exist 
between  unlike  numbers ;  and  quantities  that  may  be  expressed  in 
different  denominations  of  the  same  unit  value  must  be  reduced  to 
the  same  denomination  before  a  ratio  can  exist. 

983.  The  terms  of  a  ratio  are  the  numbers  compared.      Taken 
together  the  terms  of  a  ratio  are  sometimes  called  a  couplet. 

984.  The  first  term  of  a  ratio  is  sometimes  called  the  antecedent, 
and  the  second  term  the  consequent. 

985.  General  Principles.     Since  the  antecedent  corresponds  to  the 
dividend  in  a  question  in  division  or  the  numerator  of  a  fractional 
form,  and  the  consequent  to  the  divisor  or  the  denominator  of  a 
fractional  form  it  follows  : 

1.  That  any  change  in  the  antecedent  produces  a  like  change  in 
the  value  of  the  ratio. 

2.  That   any  change   in  the   consequent   produces  an  opposite 
change  in  the  value  of  the  ratio. 

3.  That  a  like  change  in  both  antecedent  and  consequent  does 
not  affect  the  value  of  the  ratio. 

986.  Ratios  may  be  either  direct  or  inverse. 

987.  A  direct  ratio  is  the  quotient  of  the  antecedent  divided  by 
the  consequent.     An  inverse  ratio  is  the  reciprocal  of  a  direct  ratio, 
or  the  quotient  of  the  consequent  divided  by  the  antecedent. 

Thus,  the  direct  ratio  of  8 :  4  is  8  -f-  4,  or  2  ;  and  the  inverse  ratio  of  8 : 4  is 
4  -4-  8,  or  |,  equal  to  J,  the  reciprocal  of  2. 

377 


378  RATIO   AND   PROPORTION  [§§988-995 

988.  A  simple  ratio  is  the  ratio  of  two  numbers.     A  compound 
ratio  is  the  ratio  of  the  products  of  the  corresponding  terms  of  two 
or  more  ratios. 

PROPORTION 

989.  Proportion  is  an  equality  of  ratios.   It  is  indicated  by  placing 
the  sign  of  equality  =,  or  a  double  colon  ::  between  the  ratios. 

990.  The  first  and  fourth  terms  of  a  proportion  are  called  the 
extremes,  and  the  second  and  third  terms,  the  means. 

991.  The  test  of  proportion  is  that  the  two  integers  or  fractions 
representing  the  ratios  are  equal,  or  that  the  product  of  the  extremes 
is  equal  to  the  product  of  the  means. 

992.  Hence,  a  missing  extreme  may  be  found  by  dividing  the 

product  of  the  means  by  the  given  extreme ;  and  a  missing  mean 
may  be  found  by  dividing  the  product  of  the  extremes  by  the  given 
mean. 

ORAL  EXERCISE 

Find  the  unknown  term  in  each  of  the  following  proportions : 
1.   16:4::?:12.  8.   ?:8::20:4.  5.   12:2::18:? 

8.  30:?::25:&  4.  12:?::75:25.  6.  6:8::24:? 

SIMPLE  PROPORTION 

993.  A  simple  proportion  is  an  equality  of  two  simple  ratios. 

994.  In  problems  in  simple  proportion  two  ratios  are  given,  one 
having  both  terms  and  the  other  having  only  one  term.     In  order  to 
find  the  missing  term  two  of  the  given  terms  must  be  of  one  kind, 
and  the  answer  and  the  other  term  of  another  kind. 

995.  Example.     If  8  hats  cost  $32,  what  will  11  hats  cost  at  the 

same  rate? 

SOLUTION 

For  convenience  write  ?  for  the  required  or  fourth  term. 

The  third  and  fourth  terms  must  be  of  the  same  kind.     Therefore 

$32  is  the  third  term. 

11  hats  will  cost  more  than  8  hats.     Therefore 

8  hats  is  the  first  term  and  11  hats  the  second  terra. 


§§  99-5-998]  PROPORTION  379 

The  proportion  may  then  be  stated  as  follows  : 
8  hats:  11  hats::  $32  :? 
8  hats  :  11  hats  is  equivalent  to  8  :  1 1.     Therefore 

The  required  cost  equals  ®'*2  x  11  or  $44. 
8 

996.  From  the  foregoing  explanation  the  following  rule  may  be 
derived : 

For  the  third  term  of  the  proportion  write  the  given  num- 
ber that  is  of  the  same  kind  as  the  required  fourth  term. 

From  the  nature  of  the  question  determine  whether  the 
answer  is  to  be  greater  or  less  than  the  third  term.  If  greater, 
write  the  larger  of  tli^e  two  remaining  numbers  for  the  second 
term  and  the  smaller  for  the  first  term;  if  less,  write  the 
smaller  of  the  two  remaining  numbers  for  the  second  term 
and  the  larger  for  the  -first  term. 

Divide  the  product  of  the  means  by  the  given  extreme, 
and  the  result  is  the  required  answer. 

WRITTEN  EXERCISE 

1.  If  a  post  1\  ft.  high  casts  a  shadow  \\  ft.  long,  what  is  the 
height  of  a  tower  that  casts  a  shadow  150  ft.  at  the  same  time  ? 

2.  If  15  bushels  of  wheat  can  be  bought  for  f  13.50,  how  many 
bushels  can  be  bought  for  $430.20  ? 

8.  An  insolvent  debtor  owes  $  14,400,  ana  has  an  estate  valued 
at  $10,800.  How  much  will  A  receive  on  a  claim  of  $  3750  ? 

4.  A  friend  loaned  me  $  750  for  3  yr.  4  mo.  15  da.     For  what 
period  of  time  should  1  loan  him  $  900  to  fully  repay  his  favor  ? 

5.  If  17  acres  of  pasture  land  are  sufficient;  for  51  sheep,  how 
many  acres  will  be  sufficient  for  420  sheep  ? 

COMPOUND  PROPORTION 

997.  A  compound  proportion  is  a  proportion  in  which  one  or  both 
the  ratios  are  compound. 

998.  A  compound  ratio  may  be  reduced  to  a  simple  ratio  by 
finding  the  product  of  all  the  antecedents  for  a  new  antecedent,  and 
the  product  of  all  the  consequents  for  a  new  consequent.     Hence  a 
statement  in  compound  proportion  may  be  simplified  in  practically 
the  same  manner  as  a  statement  in  simple  proportion. 


380  RATIO  AND  PROPORTION  [§§999-1001 

999.  Problems  in  compound  proportion  may  be  stated  in  the  same 
manner  as  outlined  for  simple  proportion,  or  a  statement  involving 
the  principle  of  cause  and  effect  may  be  used. 

1000.  Every  question   in  proportion  involves   the   principle   of 
cause  and  effect ;  that  is,  work  done  for  pay,  cash  given  for  goods, 
wood  cut  for  labor  performed,  etc. 

Hence  questions  in  proportion  may  be  stated  as  follows : 

1st  cause  :  2d  cause : :  1st  effect :  2d  effect. 

1001.  Example.     If  10  men  working  12  days  of  8  hours  each  can 
cut  200  cords  of  wood,  how  many  cords  should  be  cut  by  12  men  in 
15  days,  if  they  work  6  hours  per  day  ? 

SOLUTION 

The  1st  cause  =  10  men  working  for  12  days  of  8  hours  per  day. 

The  first  effect  produced  =  200  cords. 

The  2d  cause  =  12  men  working  for  15  days  of  6  hours  per  day. 

The  2d  effect  produced    =  what  number  of  cords  ? 

Hence  we  have  the  following  statement : 

10:12 

12  : 16 : :  200  cords  :  what  number  of  cords  ? 

8:6 

Or, 

12  x  15  x  6  x  200  cords  =  225  CQrdg> 
10  x  12  x  8 

WRITTEN    EXERCISE 

1.  If  15  men  earn  $  607.50  in  18  days,  how  much  should  21  men 
earn  in  12  days  ? 

2.  If  6  men,  working  for  12  days,  dig  a  ditch  80  rods  long,  how 
many  rods  of  such  ditch  should  15  men  dig  in  21  days  ? 

3.  If  5  men,  working  6  days  of  12  hours  per  day,  can  cut  24 
acres  of  corn,  how  many  acres  of  corn  should  8  men  cut  in  5  days, 
if  they  work  10  hours  per  day  ? 

4.  If  $145.35  interest  accrue  on  $510,  at  6%,  in  4  yr.  9  mo., 
how   much   interest  will    accrue   at  the   same   rate   and   time   on 
$1350? 

5.  If  $760,  put  at  interest  at  10%,  accrue  $9.50  interest  in  45 
days,  in  how  many  days  will  $1140  accrue  $17.67  interest  at  6%  ? 


STORAGE 

1002.  Storage  is  a  charge  made  for  storing  movable  property  in  a 
warehouse. 

1003.  The  rates  of  storage  may  be  fixed  by  an  agreement  between 
the  parties  to  a  contract,  but  they  are  often  regulated  by  boards  of 
trade,  chambers  of  commerce,  associations  of  warehousemen,  and  by 
legislative  enactment. 

1004.  The  term  of  storage  is  the  period  of  time  for  which  storage 
charges  are  made. 

1005.  Storage  may  be  charged  at  a  fixed  rate  per  package,  etc., 
or  at  a  fixed  sum  per  term ;  but  it  is  usually  charged  by  the  day, 
week,  or  month,  and  a  fractional  term  is  counted  as  a  full  term. 

CASH  STORAGE 

1006.  Cash  storage  is  the  term  applied  to  cases  in  which  the 
storage  is  paid  or  estimated  at  the  time  of  the  withdrawal  of  goods 
from  the  warehouse. 

1007.  In  cash  storage  all  goods  delivered  are  deducted  from  the 
oldest  receipt  on  hand. 

In  private  bonded  warehouses  of  the  United  States  goods  may  be  taken  out 
at  any  time  in  quantities  not  less  than  an  entire  package,  or  if  in  bulk,  of  not 
less  than  one  ton,  by  the  payment  of  duties,  storage,  and  labor  charges.  The 
storage  charges  are  computed  for  periods  of  one  month  each,  a  fractional  part  of 
a  month  being  counted  the  same  as  a  full  month. 

1008.  Example.    At  5  ^  per  barrel  per  month,  or  fraction  thereof, 
how  much  should  be  paid  for  the  following  storage  of  apples,  storage 
being  charged  at  each  delivery  ? 

Receipts  Deliveries 

Sept.  1,  2000  bbl.          Oct.  20,  1000  bbl. 
Nov.  20,  300  bbl.          Nov.  2,  300  bbl. 

Dec.  20,  1000  bbl. 
381 


382  STORAGE  [§  1008 

SOLUTION 

Date  Receipts  and  Deliveries  Bate          Storage 

Sept.  1  received  2000  bbl. 
Oct.  20  delivered  1000  bbl.,  which  were  in  storage  49  da.  10  <*  $  100 

1000  bbl.,  balance  in  storage 
Nov.  2  delivered    300  bbl.,  which  were  in  storage  62  da.  16  J*  45 

700  bbl.,  balance  in  storage 
Nov.  20  received    300  bbl. 

1000  bbl.,  balance  in  storage 

Dec.  20  delivered  1000  bbl.,  700  of  which  were  in  storage  110  da.    20  ^  140 

300  of  which  were  in  storage  30  da.  6  ^  15 

Total  storage  $300 

WRITTEN  EXERCISE 

1.  Find  the  storage  of  the  following  at  2  ^  per  month,  or  fraction 
thereof,  storage  being  calculated  at  each  delivery. 

Receipts  Deliveries 

Sept.  5,  200  cases  Oct.  1,  100  cases 

Sept.  30,  400  cases  Nov.  2,  200  cases 

Nov.  1,  100  cases  Dec.  1,  200  cases 

Nov.  8,  200  cases  Dec.  10,  400  cases 

2.  At  a  warehouse  there  was  received  and  delivered  flour  as 

follows :  Receipts  Deliveries 

Jan.  3,  150  bbl.  Jan.  23,  250  bbl. 

Jan.  20,  200  bbl.  Mar.  1,  400  bbl. 

Feb.  1,  300  bbl. 

The  storage  charge  on  the  above  was  5^  per  barrel  for  the  first 
10  days,  or  fraction  thereof,  and  3^  per  barrel  for  each  subsequent 
period  of  10  days,  or  fraction  thereof.  What  sum  must  be  paid  in 
settlement  ? 

3.  The  receipts  and  deliveries  of  goods  at  a  storage  warehouse 
were  as  follows : 

Receipts  Deliveries 

Sept.  2,  100  bbl.  Sept.  20,  100  bbl. 

25,  200  bbl.  30,  100  bbl. 

Oct.  19,  350  bbl.  Oct.  10,  100  bbl. 

31,  150  bbl.  20,  100  bbl. 

Nov.  7,  200  bbl.  30,  100  bbl. 

Nov.  20,  500  bbl. 


§§  1008-1010]  CREDIT   OR   AVERAGE    STORAGE  383 

The  contract  required  the  payment  of  6^  per  barrel  for  the 
present  term  of  30  days,  or  fraction  thereof,  and  3j  per  barrel  for 
each  subsequent  term  of  30  days,  or  fraction  thereof.  Find  the 
storage  bill. 

CREDIT  OR  AVERAGE  STORAGE 

1009.  Credit  or  average  storage  is  the  term  applied  to  cases  in 
which  the  storage  is  not  paid  or  estimated  until  the  last  withdrawal 
is  made. 

1010.  Example.     The  storage  charges   being  2^  per  barrel  per 
month  of  30  days,  what  will  be  the  bill  in  the  following  transaction  ? 

Receipts  Deliveries 

July  19,  200  bbl.  Aug.  15,  100  bbl. 

31,  300  bbl.  Sept.  17,  400  bbL 

SOLUTION 

From  July  19  to  July  31  =  12  da.,  200  bbl.  stored  for  12  da. 

=  1  bbl.  stored  for  2,400  da. 
July  31  300  bbl.  received. 

From  July  31  to  Aug.  15  =  15  da.,  600  bbl.  stored  for  15  da. 

=  1  bbl.  stored  for  7,500  da. 
Aug.  15  100  bbl.  delivered. 

From  Aug.  15  to  Sept.  17  =  33  da.,  400  bbl.  stored  for  33  da. 

=  1  bbl.  stored  for  13,200  da. 
Sept.  17  400  bbl.  delivered. 

Total,  1  bbl.  stored  for  23,100  da. 

28,100  -v-  30  =  770  terms  of  30  da.  each. 
$.02  x  770  =  $15.40,  the  total  storage  bilL 

WRITTEN  EXERCISE 

1.  There  was  received  at  a  warehouse :  May  30,  4000  bu.  wheat ; 
June  5,  2600  bu.  oats ;  June  24,  3500  bu.  barley ;  July  18,  5000  bu. 
corn.     If  all  of  this  was  shipped  July  20,  what  was  the  storage  bill, 
the  charge  being  1-|^  per  bushel  per  term   of  30   days7   average 
storage  ? 

2.  What  will  be  the  storage  charge,  at  4-J  ^  per  barrel,  for  a  term 
of  30  days'  average,  in  the  following  transaction  ? 


384  STORAGE  [§  1010 

Receipts  Deliveries 

Feb.     8,  180  bbl.  flour  Mar.    1,  100  bbl.  apples 

27,  100  bbl.  apples  28,  190  bbl.  flour 

Mar.    8,    60  bbl.  potatoes         Apr.  15,    60  bbl.  potatoes 
13,  300  bbl.  flour  15,    60  bbl.  flour 

29,  230  bbl.  flour 

8.  A  farmer  received  for  pasture:  Apr.  30,  12  head  of  cattle-, 
May  15,  14  head  of  cattle;  May  23,  27  head  of  cattle;  June  9, 
5  head  of  cattle;  June  30,  8  head  of  cattle;  July  16,  40  head  of 
cattle.  All  were  delivered  July  25,  and  the  charges  were  75^  per 
head  for  each  week  of  7  days'  average  pasturage.  How  much  was 
his  bill  ? 

4.  A  drover  hired  a  pasture  of  a  farmer,  agreeing  to  pay  $4.20 
per  head  of  stock  pastured  for  each  average  term  of  30  days.     What 
was  the  amount  of  the  bill,  the  receipts  and  deliveries  being  as 
follows  ? 

Receipts  Deliveries 

June  15,  21  head  of  cattle  July    1,  30  head  of  cattle 

27,  20  head  of  cattle  20,  15  head  of  cattle 

July    5,  15  head  of  cattle  30,  15  head  of  cattle 

29,  40  head  of  cattle  Aug.  21,  the  remainder 
31,  40  head  of  cattle 

5.  Find  the  storage  charges,  at  3^  per  barrel,  for  a  term  of  30 
days'  average,  on  the  following: 

Receipts  Deliveries 

Sept.  2,  1620  bbl.  Sept.  13,  520  bbl. 

16,  2920  bbl.  26,  966  bbl. 

Oct.  25, 1470  bbl.  Dec.  2,  4524  bbl. 


APPENDIX 

METRIC  SYSTEM  OF  MEASURES 

1011.  The   metric   system  is  a  decimal   system   of    denominate 
numbers.     It  is  in  use  in  nearly  all  the  European  states,  in  South 
America,  Mexico,  and  Egypt.     It  is  also  used  in  parts  of  Asia,  is 
authorized  by  law  in  the  United  States,  and  is  almost  universally 
used  in  scientific  treatises. 

1012.  The  fundamental  unit  of  the  metric  system  is  the  meter,  a 

measure  of  length,  which  is  equal  to  about  one  ten-millionth  of  the 
distance  from  the  equator  to  the  pole.  It  is  defined  by  law  as  being 
the  length  of  the  bar  of  platinum  which  is  carefully  preserved  at 
Paris. 

Accurate  copies  of  the  meter  have  been  procured  by  the  governments  of  all 
civilized  nations. 

1013.  Among  the  advantages  claimed  for  the  metric  system  are: 

1.  It  employs  only  five  unit  words  and  seven  prefixes. 

2.  Every  word  used  suggests  its  measure. 

3.  It  is  consistent,  uniform,  simple,  and  complete,  and  would  do 
away  with  the  present  inconsistent  system  of  compound  numbers. 

4.  Being  a  decimal  system  it  makes  arithmetical  operations  relat- 
ing to  measure  much  more  simple. 

5.  It  gives  to  the  nations  a  uniform  system  of  measures  and  thus 
materially  facilitates  trade  and  exchange. 

1014.  The  principal  units  of  the  metric  system  are : 

1.  The  meter  for  lengths. 

2.  The  square  meter  for  small  surfaces  such  as  floors,  ceilings,  etc. 

3.  The  are,  of  100  square  meters,  for  large  surfaces  such  as  land 
measurements. 

4.  The  cubic  meter  for  solids. 

5.  The  liter  for  capacities. 

6.  The  gram  for  weights. 

386. 


APPENDIX 


[§§  1015-1018 


1015.  Each  of  the  metric  units  is  divided  and  multiplied  deci- 
mally.    The  higher  orders  are  indicated  by  four  Greek  prefixes,  as 
follows  :  deka,  meaning  10 ;  hecto,  meaning  100 ;  kilo,  meaning  1000 ; 
myria,  meaning  10,000.     The  lower  orders  are  indicated  by  three 
Latin   prefixes,  as   follows :    deti,  meaning  .1  j   centi,  meaning  .01  j 
milli,  meaning  .001. 

Metric  Long  Measure 

1016.  The  unit  of  long  measure  is  the  meter. 

TABLE 


10  millimeters  (mm.) 
10  centimeters 
10  decimeters 
10  meters 
10  dekameters 
10  hektometers 
10  kilometers 

= 

1 
1 
1 

1 
1 
1 
1 

centimeter    (cm.)  = 
decimeter     (dm.)  = 
meter             (m.)  = 
dekameter  (Dm.)  •= 
hektometer  (Hm.)  = 
kilometer     (Km.)  = 
myriameter  (Mm.)  = 

.01 
.1 
1. 
10. 
100. 
1000. 
10,000. 

meter, 
meter, 
meter, 
meters, 
meters, 
meters, 
meters. 

In  the  above  and  each  of  the  following  tables  the  units  in  common  use  are 
indicated  by  the  blackfaced  type. 

In  the  metric  system  all  abbreviations  which  indicate  a  fractional  part  of  a 
standard  unit  begin  with  a  small  letter,  while  all  those  which  indicate  a  multiple 
of  a  standard  unit  begin  with  a  capital  letter. 


Metric  Square  Measure 

1017.  The  units  of  square  measure  are  the  square  meter  for  small 
surfaces  and  the  are  for  land  measurements. 

1018.  The  units  of  square  measure  are  the  square  of  the  units 
of  long  measure ;  hence  100  units  of  any  given  denomination  are 
required  for  1  of  the  next  higher. 


100  sq.  millimeters 
100  sq.  centimeters 
100  sq.  decimeters 
100  sq.  meters 


TABLE 

=  1  sq.  centimeter  (sq.  cm.)  = 
=  1  sq.  decimeter  (sq.  dm.)  = 
=  1  sq.  meter  (sq.  m.)  • 

=  1  sq.  dekameter  (sq.  Dm.)  = 


100  sq.  dekameters    =  1  sq.  hektometer  (sq.  Hm.)  =      10.000. 
100  sq.  hektometers   =  1  sq.  kilometer  (sq.  Km.)  =  1,000,000. 


.0001  sq.  meter. 
.01 

1.  meter  =  1  centare. 

100.          meters  =  1  are. 

"    =  1  hectare. 


§§  1010-1024]  METRIC  SYSTEM  OF   MEASURES  887 

1019.  The  centare,  the  are,  and  the  hectare,  are  used  only  in 
measuring  land. 

Metric  Cubic  Measure 

1020.  The  units  of  cubic  measure  are  the  cubic  meter,  for  ordinary 
solids,  and  the  stere,  for  wood  measurements. 

1021.  The  units  of  cubic  measure  are  the  cube  of  the  units  of 
long  measure;   hence  1000  units  of  any  given  denomination  are 
required  for  1  of  the  next  higher. 

TABLE 

1000  cu.  millimeters  (cu.  mm.)     «  1  cu.  centimeter  (en.  cm.)      ••   .000001  cu.  meter. 
1000  cu.  centimeters  —  1  cu.  decimeter  (cu.  dm.)        =    .001        '*       " 

1000  cu.  decimeters  «=  1  cu.  meter  (cu.  m.)  — 1.  ••      " 

Metric  Measures  of  Capacity 

1022.  The  unit  of  capacity,  for  both  liquid  and  dry  measures,  is 
the  liter. 

1023.  The  liter  is  a  cube  whose  side  is  1  decimeter;   hence  a 

liter  is  a  cubic  decimeter. 

TABLE 

=        .01  liter. 
=        .1     " 
=       1.       " 
=     10.     liters 
=    100.       " 
=  1,000.      " 

Measures  of  Weight 

1024.  The  unit  of  weight  is  the  gram,  which  is  equal  to  1  cubic 
centimeter  of  pure  water  at  its  greatest  density ;  that  is,  at  a  tem- 
perature just  above  freezing. 

TABLE 

10  milligrams  (mg.)  =  1  centigram  (eg.)  =  .01  gram. 

10  centigrams  =  1  decigram  (dg.)  =  .1       *' 

10  decigrams  =  1  gram  (g.)  =  1.        " 

10  grams  =  1  dekagram   (Dg.)  =  10.  grams. 

10  dekagrams  =  1  hektogram  (Hg.)  =  100.         «« 

10  hektograms  =  1  kilogram  (Kg.)  =  1,000.        •« 

10  kilograms  =  1  myriagram  (Mg.)  =  10,000.        •• 

10  myriasrams  =  1  quintal  (Q. )  =  100,000.        •« 

10  quintals  =  1  tonneau  (T.)  =  100,0000.  "«« 


10  milliliters  (ml.) 
10  centiliters 
10  deciliters 
10  liters 
10  dekaliters 
10  hektoliters 

=  1  centiliter  (cl.) 
=  1  deciliter  (dl.) 
=  1  liter  (1.) 
=  1  dekaliter  (Dl.) 
=  1  hektoliter  (HI.) 
=  1  kiloliter  (Kl.) 

.  338  APPENDIX  [§  1025 

1025.  An  act  of  Congress  requires  all  reductions  from  the  metric 
to  the  common  system,  or  the  reverse,  to  be  made  according  to  the 
following 

TABLE  OF  EQUIVALENTS 
Long  Measure 

1  inch  =  2.54  centimeters.  1  centimeter  =    .3937  of  an  inch. 

1  foot  =    .3048  of  a  meter.  1  decimeter  =    .328  of  a  foot. 

1  yard  =    .9144  of  a  meter.  1  meter          =  1.0936  yards. 

1  rod    =  5.029  meters.  1  dekameter  =  1.9884  rods. 

1  mile  =  1.6093  kilometers.  1  kilometer   =    .62137  of  a  mile. 

Square  Measure 

1  sq.  inch  =    6.452  sq.  centimeters.  1  sq.  centimeter  =    .155  of  a  sq.  inch. 

1  sq.  foot  =      .0929  of  a  sq.  meter.  1  sq.  decimeter  =    .1076  of  a  sq.  foot. 

1  sq.  yard  =      .8361  of  a  sq.  meter.  1  sq.  meter          =  1.196  sq.  yards. 

1  sq.  rod    =  25.293  sq.  meters.  1  are  =  3.954  sq.  rods. 

1  acre        =40.47  ares.  1  hektare  =  2.471  acres. 

1  sq.  mile  =  259  hectares.  1  sq.  kilometer   =    .3861  of  a  sq.  mile. 

Cubic  Measure 

I  cu.  inch  =  16.387  cu.  centimeters.  1  cu.  centimeter  =    .061  of  a  cu.  inch. 

1  cu.  foot  =  28.317  cu.  decimeters.  1  cu.  decimeter  =    .0353  of  a  cu.  foot 

1  cu.  yard  =     .7645  of  a  cu.  meter.  1  cu.  meter         =  1.308  cu.  yards. 

1  cord        =    3.624  steres.  1  stere  =    .2759  of  a  cord. 


Measures  of  Capacity 

1  liquid  quart  =    .9463  of  a  liter.  1  liter  =  1.0567  liquid  quarts. 

1  dry  quart       =  1.101  liters.  1  liter  =    .908  of  a  dry  quart. 

1  liquid  gallon  =    .3785  of  a  dekaliter.  1  dekaliter  =  2.6417  liquid  gallons. 

1  peck  =    .881  of  a  dekaliter.  1  dekaliter  =  1.135  pecks. 

1  bushel  =    .3524  of  a  hektoliter.  1  hektoliter  =  2.8375  bushels. 


Measures  of  Weight 

1  grain,  Troy     =      .0648  of  a  gram.  1  gram       =     .03527  of  an  ounce,  Avoir. 

1  ounce,  Avoir.  =  28.35  grams.  1  gram       =     .03215  of  an  ounce,  Troy. 

1  ounce,  Troy    =  31.104  grams.  1  gram       =15.432  grains,  Troy. 

1  pound,  Avoir.  =      .4536  of  a  kilogram.  1  kilogram  =  2.2046  pounds,  Avoir. 

1  pound,  Troy    =      .3732  of  a  kilogram.  1  kilogram  =  2.679  pounds,  Troy. 

I  ton  (short)      =     .9072  of  a  tonneau.  1  tonneau  =  1.1023  tons  (short). 


§§  1025-1028]  POWERS   AND   ROOTS  389 

WRITTEN  EXERCISE 

1.  What  will  be  the  cost  in.  Paris  of  a  cargo  of  38,500  bu.  United 
States  wheat  at  10  francs  60  centimes  per  hektoliter  ? 

2.  How  many  avoirdupois  pounds  in  10  myriagrams  4  kilo- 
grams ? 

3.  Reduce  250  hectares  to  common  units. 

4.  A  pile  of  wood  56  meters  long,  18J  meters  wide,  and  3f 
meters  high  was  sold  at  $6  per  cord.     How 'much  was  received 
for  it? 

5.  At  21^  per  liter,  what  will  150  quarts  of  olive  oil  cost  ? 

6.  If  the  cost  of  50  liters  of  wine  was  800  francs,  what  was  the 
price  per  gallon  in  United  States  money  ? 

7.  A  merchant  bought  silk  at  $  1.20  per  meter  and  sold  it  by 
the  yard  at  a  gain  of  20%.     What  was  the  selling  price  per  yard  ? 

8.  A  man  bought  50  kilograms  of  sugar  for  $  5.51  and  sold  it 
at  a  gain  of  20%.     What  did  he  receive  per  pound  ? 

9.  A  pile  of  wood  8  meters  long  and  2  meters  wide  contains 
56  steres.     Find  the  height  of  the  pile  in  meters ;  in  feet  and  inches. 

10.  Reduce  954  miles  to  kilometers. 

11.  How  many  meters  of  carpet  70  centimeters  wide  will  be 
required  for  a  room  7  meters  long  and  5  meters  wide,  if  the  strips 
run  crosswise  and  4  meters  are  lost  in  matching  the  pattern  ?   how 
many  yards  ? 

12.  How  many  hectares  in  a  field  150  meters  on  a  side  ?   how- 
many  acres  ? 

POWERS   AND   ROOTS 

1026.  The  power  of  a  number  is  the  product  arising  from  multi- 
plying the  number  by  itself  one  or  more  times. 

1027.  A  perfect  power  is  a  number  that  can  be  exactly  produced 
by  the  involution  of  some  number  as  a  root. 

Thus,  25  and  8  are  perfect  powers,  since  5  x  5  =  25,  and  2x2x2  =  8. 

1028    The  square  of  a  number  is  its  second  power. 


390  APPENDIX 

1029.  The  cube  of  a  number  is  its  third  power. 


[§§  1029-1033 


1030.  An  exponent  is  a  small  figure  written  at  the  right  of  a 
number  to  indicate  how  many  times  the  number  is  to  be  used  as  a 
factor. 

Thus,  22  is  equivalent  to  2  x  2  and  is  read  the  second  power  of  2  ;  and  53  is 
equivalent  to  5  x  5  x  5  and  is  read,  'the  third  power  of  5. 

1031.  The  root  of  a  number  is  one  of  the  equal  factors  which 
multiplied  together  will  produce  the  given  number. 


SQUARE  ROOT 

1032.   The  square  root  of  a  number  is  one  of  the  two  equal  factors 
which  multiplied  together  will  produce  the  given  number. 

The  accompanying  diagram  is  a 
square  14  ft.  on  a  side.  Its  area  is, 
by  inspection,  found  to  be  made  up  of : 

1.  The  tens  of  14,  or  102,  equal  to 
100  sq.  ft.,  as  shown  by  the  square 
within  the  angles,  a,  £>,  c,  d. 

2.  Twice  the  product  of  the  tens 
by  the  units  of  the  same  number  or  2  x 
(10  x  4),  equal  to  80  sq.  ft.,  as  shown 
by  the  surface  within  the  angles  e,  /, 
gr,  h  and  z,  j,  k,  I. 

3.  The  square  of  the"  units,  4  ft, 
equal  to  16  sq.  ft.,  as  shown  by  the 
square  within  the  angles  to,  x,  y,  z. 


a 

d 

c 

h 

b 

c 

f 

(J 

i 

I 

w 

z 

— 

J 

k 

X 

y 

14  ft.  =  10  ft.  and  4ft. 


Hence  a  square  14  ft.  on  each  side  will  contain  : 

102  =  100  sq.  ft. 

2  x  (10  x  4)        =    80  sq.  ft. 

42  =    16  sq.  ft. 

142  =  196  sq.  ft. 

1033.   Therefore  the  following  general  principle  may  be  stated : 

The  square  of  any  number,  composed  of  two  or  more  figures, 
is  equal  to  the  square  of  the  tens  plus  twice  the  product  of  the 
tens  multiplied  by  tlw  units  plus  the  square  of  the  units. 


§§  1034-1036]  POWERS   AND  ROOTS  391 

1034.  In  extracting  the  square  root  of  a  number,  the  first  impor- 
tant step  is  to  separate  the  figures  of  which  the  number  is  composed 
into  groups. 

The  squares  of  1,  2,  3,    4,    5,    6,    7,    8,    9,    10. 
are  1,  4,  9,  16,  25,  36,  49,  64,  81,  100. 

From  the  above  it  is  evident : 

1.  That  the  square  of  any  number  will  contain  at  least  one  place  or  one 
order  of  units. 

2.  That  the  square  of  no  number  represented  by  a  single  figure  will  contain 
more  than  two  places. 

3.  That  if  the  number  of  which  the  square  root  is  sought  be  separated  into 
periods  of  two  figures  each,  beginning  at  the  units,  the  number  of  periods  and 
partial  periods  so  made  will  represent  the  number  of  unit  orders  in  the  root. 

4.  That  the  square  of  any  number  will  contain  twice  as  many  places  or  one 
less  than  twice  as  many  places  as  its  root. 

5.  That  where  the  product  of  the  left-hand  figure  multiplied  by  itself  is  not 
greater  than  9,  then  the  square  will  contain  one  less  than  twice  as  many  places 
as  the  root. 

1035.  Example.    Find  the  square  root  of  625. 

6  OK /OK  SOLUTION.  The  number  consists  of  one  full  and  one 

partial  period  ;  hence,  its  root  will  contain  two  places. 

— The  given  number,  625,  is  the  second  power  of  the  root  to 

45)2  25  be  extracted  ;  therefore  the  first  figure  of  the  root,  which  will 

2  25  be  the  highest  order  of  units  in  that  root,  must  be  obtained 

from  the  first  left-hand  period.  The  first,  or  tens'  figure, 
of  the  root  will  be  the  square  root  of  the  greatest  perfect  square  in  6.  Hence,  2 
is  the  tens'  figure  of  the  root.  Subtracting  the  tens,  the  remainder,  225,  must  be 
equal  to  twice  the  tens  multiplied  by  the  units  plus  the  square  of  the  units. 
Twice  the  2  tens  is  equal  to  4  tens.  4  tens  is  contained  in  the  22  tens  of  the 
remainder  5  times  ;  hence,  5  is  the  units'  figure  of  the  root.  Twice  the  tens  mul- 
tiplied by  the  units  plus  the  square  of  the  units  is  equivalent  to  twice  the  tens 
plus  the  units  multiplied  by  the  units.  Therefore,  5  units  are  annexed  to  the  4 
tens  and  the  result,  45,  is  multiplied  by  5.  Therefore,  the  square  root  of  625 
is  25. 

1036.  From  the  foregoing  explanations  the  following  rule  may 
be  derived : 

Beginning  at  the  right,  separate  the  given  number  into 
periods  of  two  places  each. 

Take  the  square  root  of  the  greatest  perfect  square  con- 
tained in  the  left-hand  period  for  the  first  root  figure;  sub- 


392  APPENDIX  [§§  1036-1037 

tract  its  square  from,  the  left-hand  period,  and  to  the 
remainder  bring  down  the  next  period. 

Divide  the  number  thus  obtained,  exclusive  of  its  units, 
by  twice  the  root  figure  already  found  for  a  second  quotient 
or  root  figure.  Place  this  figure  at  the  right  of  the  root  figure 
before  found,  and  also  at  the  right  of  the  divisor. 

Multiply  the  divisor  thus  formed  by  the  new  root  figure. 
Subtract  the  result  from  the  dividend,  to  the  remainder 
bring  down  the  next  period,  and  so  proceed  until  the  last 
period  has  been  brought  down,  considering  the  entire  root 
already  found  as  so  many  tens  in  determining  subsequent 
root  figures. 

Whenever  the  divisor  is  greater  than  the  dividend,  place  a  cipher  in  the  root, 
and  also  at  the  right  of  the  divisor  ;  bring  down  another  period,  and  proceed  as 
before. 

When  the  root  of  a  mixed  decimal  is  required,  form  periods  from  the  decimal 
point  right  and  left,  and  if  necessary  supply  a  decimal  cipher  to  make  the  decimal 
periods  of  two  places  each. 

Any  root  of  a  common  fraction  may  be  obtained  by  extracting  the  root  of 
the  numerator  for  a  numerator  of  the  root,  and  the  root  of  the  denominator  for 
the  denominator  of  the  root. 

To  find  a  root,  decimally  expressed,  of  any  common  fraction,  reduce  such 
fraction  to  a  decimal,  and  extract  the  root  to  any  number  of  places. 


WRITTEN  EXERCISE 
Find  the  square  root  of : 

1.  196.  5.  5625.  9.   125.44. 

2.  225.  6.   42436.  10.   50.2681. 

3.  576.  7.   15625.  11. 

4.  1225.  8.  1048576.  12. 

APPLICATIONS  OF  SQUARE  ROOT 

1037.   It  has  been  shown  that  the  area  of  a  square  is  the  product 
of  its  two  equal  sides.     Hence, 

The  side  of  any  square  is  the  square  root  of  its  area. 


§§  1038-1039]  POWERS  AND   ROOTS  393 

1038.  The   hypothenuse  of  a  right-angled  triangle  is  the   side 
opposite  the  right  angle. 

1039.  The  square  formed  on  the  hypothenuse  of  a  right-angled 
triangle    is    equal    to    the    sum    of    the 

squares  formed  on  the  base  and  perpen- 
dicular.    Hence, 

The  hypothenuse  of  a  right-angled  tri- 
angle is  the  square  root  of  the  sum  of  the 
squares  of  the  other  two  sides;  and 

The  base  or  perpendicular  of  a  right- 
angled  triangle  is  the  square  root  of  the  dif- 
ference between  the  square  of  the  hypothe- 
nuse  and  that  of  the  given  side. 

WRITTEN  EXERCISE 

1.  The  base  of  a  figure  is  60  ft.  and  the  perpendicular  80  ft. 
What  is  the  hypothenuse  ? 

2.  A  farm  of  80  acres  is  in  the  form  of  a  rectangle,  the  length  of 
which  is  twice  its  width.     How  many  rods  of  fence  will  inclose  it  ? 

3.  How  many  rods  of  fence  will  inclose  a  triangular  field  whose 
base  is  equal  to  its  perpendicular  and  whose  area  is  20  acres  ? 

4.  If  a  farm  is  1  mile  square,  how  far  is  it  diagonally  across 
from   corner  to  corner?      Express   the  result  in  rods,   feet,  and 
inches. 

5.  What  is  the  width  of  a  street  in  which  a  ladder  60  ft.  long  can 
so  be  placed  that  -it  will  reach  the  eaves  of  a  building  40  ft.  high  on 
one  side  of  the  street,  and  of  another  building  50  ft.  high  on  the 
opposite  side  of  the  street  ? 

6.  How  many  feet  of  fence  will  inclose  a  square  field  containing 
16  acres  ? 

7.  How  far  apart  are  the  opposite  corners  of  a  rectangular  field 
having  a  width  equal  to  f  of  its  length  and  containing  30  acres  ? 

8.  What  is  the  length  of  one  side  of  a  square  field,  the  area  of 
which  is  one  acre  ? 


394  APPENDIX  [§§  1040-1043 

CUBE  ROOT 

1040.  The  cube  root  of  a  number  is  one  of  the  three  equal  factors 
which  multiplied  together  will  produce  the  given  number. 

Thus,  a  cubic  foot  equals  12  x  12  x  12,  or  1728  cubic  inches,  the  product  of 
its  length,  breadth,  and  thickness ;  and  since  12  is  one  of  the  equal  factors 
of  1728,  it  must  be  its  cube  root. 

1041.  The  first  point  to  be  settled  in  extracting  any  root  is  the 
relative  number  of  unit  orders  or  places  in  the  number  and  its  root. 

The  cubes  of  1, 2,    3,    4,      5,     6,      7,      8,     9,      10 
are  1,  8,  27,  64,  125,  216,  343,  512,  729,  1000 

From  the  above  it  is  evident : 

1.  That  the  cube  of  any  number  expressed  by  a  single  figure  cannot  have 
less  than  one  nor  more  than  three  places  or  unit  orders. 

2.  That  each  place  added  to  the  number  will  add  three  places  to  its  cube. 

3.  That  if  a  number  be  separated  into  periods  of  three  figures  each,  begin- 
ning at  the  units,  the  number  of  places  in  the  root  will  equal  the  number  of 
periods  and  partial  periods,  if  there  are  any. 

4.  That  the  cube  of  any  number  will  contain  three  times  as  many  places  or 
one  or  two  less  than  three  times  as  many  places  as  its  roots. 

1042.  Since  57  equal  50  -f  7,  the  cube  of  57  may  be  determined 
in  the  following  mannqr : 

50+7 
50+7 

(50 x  7)  +7*  60»=126,000  cu.  ft. 

502+(50x7)  3  x  (502  x  7)  =  52,500  cu.  ft. 

502+2  x  (50  x  7) +7*  3  x  (50  X  72)  =     7,350  cu.  ft. 

50+7  73=        343  cu.  ft. 


(50*x7)+2x(50x72)+78  673=185,193  cu.  ft. 

503+2x(502x7)  +  (50x72) 


503+3  x  (502  x  7)  +3  x  (50  x  I2)  +  7* =185,193 

1043.  Hence  the  following  general  principle  may  be  stated : 

The  cube  of  a  number  is  equal  to  the  cube  of  the  tens,  plus 
three  times  the  square  of  the  tens,  multiplied  by  the  units, 
plus  three  times  the  tens  multiplied  by  the  square  of  the 
units,  plus  the  cube  of  the  units. 


§  1044] 


POWERS  AND   ROOTS 


395 


1044.    Example.    Extract  the  cube  of  15625. 


SOLUTION.    The  given  num- 
ber consists  of  two  periods  of 

three  figures  each,  therefore  its 

3t2u-f3tu24-u3=    7625= remainder.  cube    root    wil1    contain    two 


ts+3t2u+3tu2+u3=15.625(2  5 

t3=  8         or  8000 


t2=400 
3t2=1200 
3t=     60 

3 12+3 1=1260  trial  divisor. 

3t2u=6000 

3tu2=1500 

u8=  125 

3t2u+3tu2+u3=7625. 


places. 

Since  the  given  number  is  a 
product  of  the  root  taken  three 
times  as  a  factor,  the  first  fig- 
ure, or  highest  order  of  the  root, 
must  be  obtained  from  the  first 
left-hand  period,  or  highest  or- 
der of  the  power.  The  greatest 
cube  in  15  is  8  and  the  cube 
root  of  8  is  2 ;  hence,  2  is  the 
tens'  figure  of  the  root. 
Subtracting  the  cube  of  the  root  figure  thus  found  and  bringing  down  the 

next  figure,  the  entire  remainder  is  found  to  be  7825. 

Referring  to  the  general  principle  stated  above,  we  find  that  having  sub- 
tracted from  the  given  number  the  cube  of  its  tens,  the  remainder,  7625  must 

contain  three  times  the  product  of  the  square  of  the  tens  by  the  units  plus  three 

times  the  product  of  the  tens  by  the  square  of  the  units  plus  the  cube  of 

the  units. 

If  a  cube  (^4),  20  inches  in  length  on 

each  side,  is  formed,  its  solid  contents  will 

equal  8000  cubic  inches,  and  it  will  be 

shown    that    the    remaining    7625    cubic 

inches  are  to  be  so  added  to  cube  (^4) 

that  it  will  retain  its  cubical  form.      In 

order  to  do  this,  equal  additions  must  be 

made  to  the  three   adjacent  sides;  and 

these  three  sides  being  each  20  inches  in 

length  and  20  inches  in  width,  the  addi- 
tion to  each  of  them  in  surface,  or  area,  is 

202,  and  to  the  three  sides  3  x(202),  as 

shown  in  the  squares  (U).     It  will  also  be 

observed  that  the  three  oblong  blocks,  as 

shown  in  ((7),  will  be  required  to  fill  out 

the  vacancies  in  the  edges,  and  also  the 

small  cube  (Z>),  to  fill  out  the.  corner. 
Since  each  of  the  oblong  blocks  has  a 

length  of  2  tens,  or  20  inches,  the  three 

will  have  a  length  of  3  x  20  inches.     Observe  now  that  the  surface  to  be  added 

to  the  cube  (^4),  in  order  to  include  in  its  contents  the  7625  remaining  cubic 

inches,  has  been  nearly,  but  not  exactly,  obtained ;  and  since  cubic  contents 


396 


APPENDIX 


[§§  1044-1045 


divided  by  surface  measurements  must  give  units  of  length,  the  thickness  of  the 
three  squares  (5),  and  of  the  three  oblong  pieces  (O),  will  be  determined  by 
dividing  7625  by  the  surface  of  the  three  ^ 

squares  plus  the  surface  of  the  three  oblong 
blocks.  This  division  may  give  a  quotient 
too  large,  owing  to  the  omission  in  the  divi- 
sor of  the  small  square  in  the  corner  ;  hence, 
such  surface  measure  taken  as  a  divisor  may 
with  propriety  be  called  a  trial  divisor.  So 
using  it,  5  is  obtained  as  the  second,  or  unit 
figure  of  the  root. 

Assuming  this  5  to  be  the  thickness  of 
the  three  square  blocks  (J5),  and  both  the  height  and  thickness  of  the  three 
oblong  blocks  ((7),  gives  for  the  solid  contents  of  the  three  square  blocks  (.B), 
6000,  and  for  the  solid  contents  of  the  three  oblong  blocks  ((7),  1500  ;  these  two 
added  together  equal  7500.  Again  referring  to  the  general  principle  stated 
above,  we  find  that  the  only  element  required  to  complete  the  cube  is  the  cube 
of  the  units. 

Now,  by  reference  to  the  illustrative  blocks,  observe  that  by  placing  the 
small  cube  (Z>)  in  its  place  in  the  corner,  the  cube  is  complete.  And  since  (D) 
has  been  found  to  contain  5  x  5  x  5,  or  125  cubic  inches,  add  this  sum  to  the 
7500  obtained  above,  and  the  result  is  7625.  Subtracting  7625  from  the  re- 
mainder in  the  problem,  nothing  remains  ;  hence,  it  has  been  shown  that  the 
cube  root  of  15,625  is  25.  By  the  operation  is  also  proved  the  correctness  of 
the  general  principle  as  stated. 

1045.  From  the  foregoing  explanations  the  following  rule  may 
be  derived : 

Beginning  at  the  right,  separate  the  given  number  into 
periods  of  three  figures  each. 

Take  for  the  first  root  figure  the  cube  root  of  the  greatest 
perfect  cube  in  the  left-hand  period ;  subtract  its  cube  from  this 
left-hand  period  and  to  the  remainder  bring  down  the  next 
period. 

Divide  this  remainder,  using  as  a  trial  divisor  three  times 
the-  square  of  the  root  figure  already  found,  considered  as  tens, 
so  obtaining  the  second  or  units'  figure  of  the  root ;  next  sub- 
tract from  the  remainder  three  times  the  square  of  the  tens 
multiplied  by  the  units,  plus  three  times  the  tens  multiplied 
by  the  square  of  the  units,  plus  the  cube  of  the  units. 

In  examples  of  more  than  two  periods  proceed  as  above,  and  after  two  root 
figures  are  found,  treat  both  as  tens  for  finding  the  third  root  figure.  For  finding 
subsequent  root  figures  treat  all  those  found  as  so  many  tens. 


§§  1046-1046]  POWERS   AND  ROOTS  397 

In  case  the  remainder,  at  any  time  after  bringing  down  the  next  period,  be 
less  than  the  trial  divisor,  place  a  cipher  in  the  root  and  proceed  as  before. 

Should  the  cube  root  of  a  mixed  decimal  be  required,  form  periods  from  the 
decimal  point  right  and  left.  If  the  decimal  be  pure,  point  off  from  the  decimal 
point  to  the  right,  and  if  need  be  annex  decimal  ciphers  to  make  full  periods. 

The  fourth  root  may  be  obtained  by  extracting  the  square  root  of  the  square 
root. 

The  sixth  root  is  obtained  by  taking  the  cube  root  of  the  square  root  or  the 
square  root  of  the  cube  root. 

WRITTEN  EXERCISE 
Extract  the  cube  root  of  : 

1.  1728.  4.   65939264  7.    ^il}?- 

2.  15625.  5.   T62V  8.   1264.295441. 
8.   110592.                  ft  9. 


APPLICATIONS  OF  CUBE  ROOT 

1046.   It  has  been  shown  that  the  solid  contents  of  a  cube  is  the 
product  of  its  three  equal  sides.     Hence, 

Tlie  side  of  any  cube  is  the  cube  root  of  its  solid  contents. 

WRITTEN  EXERCISE 

1.  How  many  square  inches  in  the  six  faces  of  a  cubical  block 
whose  solid  contents  are  6400  cubic  inches  ? 

2.  A  cubical  cistern  contains  3375  cubic  feet.    What  is  its  depth  ? 

8.   What  must  be  the  height  of  a  cubical  bin  that  will  hold  1000 
bushels  of  wheat  ? 

4.  A  square  cistern  the  capacity  of  which  is  420  barrels,  has  a 
depth  of  only  \  its  width.     Find  its  dimensions. 

5.  A  cubical  cistern  contains  630  barrels.     How  deep  is  it  ? 


APPENDIX 


COMPOUND  INTEREST  TABLE  FOR  ANNUAL  PAYMENTS 

Showing  how  much  $1.00  per  annum  will  amount  to,  compounded 
annually,  in  any  number  of  years  from  1  to  50  years.  (Compare 
with  pages  251,  252.) 


Tears 

3  per  ct. 

3s  per  ct. 

4  per  ct. 

4*  per  ct. 

5  per  ct. 

6  per  ct. 

Years 

1 

1.030 

1.035 

1.040 

1.045 

1.050 

1.060 

1 

2 

2.091 

2.106 

2.122 

2.137 

2.153 

2.184 

2 

3 

3.184 

3.215 

3.247 

3.278 

3.310 

3.375 

3 

4 

4.309 

4.363 

4.416 

4.471 

4.526 

4.637 

4 

5 

5.468 

5.550 

6.633 

6.717 

5.802 

5.975 

5 

6 

6.663 

6.779 

6.898 

7.019 

7.142 

7.394 

6 

7 

7.892 

8.052 

8.214 

8.380 

8.549 

8.898 

7 

8 

9.159 

9.369 

9.583 

9.802 

10.027 

10.491 

8 

9 

10.464 

10.731 

11.006 

11.288 

11.578 

12.181 

9 

10 

11.808 

12.142 

12.486 

12.841 

13.207 

13.972 

10 

11 

13.192 

13.602 

14.026 

14.464 

14.917 

15.870 

11 

12 

14.618 

15.113 

15.627 

16.160 

16.713 

17.882 

12 

13 

16.086 

16.677 

17.292 

17.932 

18.599 

20.015 

13 

14 

17.599 

18.296 

19.024 

19.784 

20.579 

22.276 

14 

15 

19.157 

19.971 

20.825 

21.719 

22.658 

24.673 

15 

16 

20.762 

21.705 

22.698 

23.742 

24.840 

27.213 

16 

17 

22.414 

23.500 

24.645 

25.855 

27.132 

29.906 

17 

18 

24.117 

25.357 

26.671 

28.064 

29.539 

32.760 

18 

19 

25.870 

27.280 

28.778 

30.371 

32.066 

35.786 

19 

20 

27.677 

29.270 

30.969 

32.783 

34.719 

38.993 

20 

21 

29.537 

31.329 

33.248 

35.303 

37.505 

42.392 

21 

22 

31.453 

33.460 

35.618 

37.937 

40.431 

45.996 

22 

23 

33.427 

35.667 

38.083 

40.689 

43.502 

49.816 

23 

24 

35.459 

37.950 

40.646 

43.565 

46.727 

63.865 

24 

25 

37.553 

40.313 

43.312 

46.571 

60.114 

68.156 

25 

26 

39.710 

42.759 

46.084 

49.711 

63.669 

62.706 

26 

27 

41.931 

45.291 

48.968 

62.993 

67.403 

67.528 

27 

28 

44.219 

47.911 

61.966 

56.423 

61.323 

72.640 

28 

29 

46.575 

50.623 

65.085 

60.007 

65.439 

78.058, 

29 

30 

49.003 

53.430 

68.328 

63.752 

69.761 

83.802 

30 

31 

61.503 

56.335 

61.702 

67.666 

74.299 

89.890 

31 

32 

54.078 

59.341 

65.210 

71.756 

79.06 

96.343 

32 

33 

56.730 

62.453 

68.858 

76.030 

84.067 

103.184 

33 

34 

69.462 

65.674 

72.652 

80.497 

89.320 

110.435 

34 

35 

62.272 

69.008 

76.598 

85.164 

94.836 

118.121 

35 

36 

65.174 

72.458 

80.702 

90.041 

100.628 

126.268 

36 

37 

68.159 

76.029 

84.970 

95.138 

106.710 

134.904 

37 

38 

71.234 

79.725 

89.409 

100.464 

113.095 

144.059 

38 

39 

74.401 

83.550 

94.026 

106.030 

119.800 

153.762 

39 

40 

77.663 

87.510 

98.827 

111.847 

126.840 

164.048 

40 

41 

81.023 

91.607 

103.820 

117.925 

134.232 

174.951 

41 

42 

84.484 

95.849 

109.012 

124.276 

141.993 

186.508 

42 

43 

88.048 

100.238 

114.413 

130.914 

150.143 

198.758 

43 

44 

91.720 

104.782 

120.029 

137.850 

158.700 

211.744 

44 

45 

95.502 

109.484 

125.871 

145.098 

167.685 

225.508 

45 

46 

99.397 

114.351 

131.945 

152.673 

177.119 

240.099 

46 

47 

103.408 

119.388 

138.263 

160.588 

187.025 

255.565 

47 

48 

107.541 

124.602 

144.834 

168.859 

197.427 

271.958 

48 

49 

111.797 

129.998 

151.667 

177.503 

208.348 

289.336 

49 

50 

116.181 

1&5.5H3 

158.774 

186.536 

219.815 

307.756 

50 

ANSWERS 

Page  15.  1.  43,400.  2.  46,987.  3.  48,099.  4.  29,273. 
5.  21,771.  6.  23,564.  7.  25,606. 

Page  19.  1.  447,136,427.  2.  387,213,476.  3.  484,730,888. 
4.  503,980,350.  5.  458,853,797.  6.  475,095,610.  7.  78,841. 
8.  84,551.  9.  69,394.  10.  79,473.  11.  76,930.  12.  53,143. 

Page  20.  13.  Vertical  totals :  34,134;  22,798;  26,057;  47,779; 
34,788;  51,426.  Horizontal  totals :  23,764;  21,751;  15,089;  26,428; 
31,480 ;  29,679 ;  30,061 ;  16,318 ;  22,412.  Total,  216,982. 

14.  Vertical  totals:  Clothing,  $4652.21;  dry  goods,  $5500.32; 
furnishings,  $849.08;  millinery,  $2357.92;  household  utensils, 
$  4011.16.  Horizontal  totals :  Monday,  $2659.05 ;  Tuesday,  $2883.10 ; 
Wednesday,  $2847.60;  Thursday,  $2613.38;  Friday,  $2937.30; 
Saturday,  $  3430.26.  Total,  $  17,370.69. 

Page  23.  1.  Vertical  totals :  Armories,  $  180,280.28 ;  metropoli- 
tan sewer,  $492,597.24;  abolition  of  grade  crossings,  $393,663.73; 
metropolitan  water,  $1,946,951.37;  highways,  $1,157.89.  Hori- 
zontal totals:  1895-1896,  $392,774.63;  1896-1897,  $404,963.27; 
1897-1898,  $  386,627.68 ;  1898-1899,  $  425,974.50 ;  1899-1900, 
$647,621.81;  1900-1901,  $756,688.62.  Total,  $3,014,650.51. 

2.  Vertical    totals:    Shoes,    $2253.49;    gloves,    $1527.23;    hats, 
$1409.44;  dress  goods,  $3002.61;  clothing,  $3211.52.     Horizontal 
totals :  A  to  D  Ledger,  $  1183.12 ;  E  to  H  Ledger,  $  1224.73 ;  I  to  L 
Ledger,  $1881.51;  M  to  P  Ledger,  $1386.58;  Q  to  T  Ledger,  $3078; 
U  to  Z  Ledger,  $2650.35.     Total,  $11,404.29. 

3.  Vertical    totals:    Domestics,    $5760.89;    notions,    $3791.25; 
woolens,    $6408.90;    dress    goods,    $4961.63.     Horizontal    totals: 
Monday,   $2056.71;    Tuesday,   $2481.41;    Wednesday,   $4749.31; 
Thursday,  $  2661.46 ;  Friday,  $  4490.56 ;  Saturday,  $  4483.22.    Total, 
$20,922.67. 

Page  24.  Vertical  totals :  Eegistered  letters,  11,420 ;  ordinary 
letters,  94,667 ;  postal  cards,  9338 ;  book  packets,  3397 ;  parcels, 


400  ANSWERS 

1516;  newspapers,  138,689.  Horizontal  totals:  Monday,  45,717; 
Tuesday,  37,584  Wednesday,  42,788;  Thursday,  47,162;  Friday, 
51,665 ;  Saturday,  34,111.  Total,  259,027. 

5.1,154,276,889.     6.662,377,884.     7.788,754,622.     8.837,865,199. 

Page  29.  1.  3380  Ib.   2.  1650  Ib.   3.  214  Ib.  4.  2520  Ib. 

Page  30.  5.  $1428.73.   6.  $7428.96.   7.  $654.97. 

Page  31.  8.  $1722.32.  9.  $409.40.  10.  $463.73.  11.  $368.15. 

Page  33.  1.  E.  W.  Allen,  $1416.84;  C.  W.  Briggs,  $814.95; 
L.  M.  Comer,  $1030.73;  0.  D.  Day,  $1477.43;  A.  L.  Emery, 
$453.81;  B.  C.  Foley,  $907.83;  J.  I.  Good,  $1087.51;  L.  O.  Hall, 
$1539.44;  Chas.  E.  Irwin,  $1929.54;  Chas.  H.  Jones,  $827.95. 
Total  new  balances,  $11,486.03.  Total  old  balances,  $9054.87. 
Total  checks,  $3391.08.  Total  deposits,  $5822.24. 

2.  D.  T.  Ames,  $  10,514.15 ;  M.  T.  Ballou,  $  7602.30 ;  W.  T.  Collins, 
$2663.89;  Dorman  &  Co.,  $2985.63;  Evans  &  Son,  $1972.10; 
Farley  Bros.,  $ 2162.01 ;  Grant  &  Snow  Co.,  $  7574.90 ;  Hall  &  Smith, 
$7578.60;  J.  T.  Irwin,  $1780.92;  M.  I.  Jamison,  $5338.33.  Total 
new  balances,  $50,172.83.  Total  old  balances,  $36,732.21.  Total 
checks,  $8232.94.  Total  deposits,  $21,673.56. 

Page  35.    1.   $2000.     2.   $4225.     3.   $755.     4.   Gained  $1638. 

5.  Lost  $96.      6.   $1056.      7.   150  A. 

Page  40.    1.  462.      2.   1584.      3.   264.      4.   14,190.      5.   14,025. 

6.  1386.  7.  4686.  8.  7062.  9.  11,000.  10.  26,708.  11.  3256. 
12.  2200.   13.  8030.   14.  4752.   15.  6644.   16.  8250. 

Page  41.  1.  1,480,016  ems.  2.  6,856,080  men.  3.  5,412,969 
links.  4.  106,272  pairs.  5.  $2,329,992.  6.  $53,816.  7.  4,557,168 
books.  8.  11,751,810  Ib.  9.  125,244  Ib.  10.  $154,716,975. 

Page  43.  1.  768.  2.  1435.  3.  67,680.  4.  2921.  5.  2226. 
6.  77,184.  7.  47,082.  8.  5220.  9.  1222.  10.  1776.  11.  30,524. 
12.  10,679.  13.  4250.  14.  2088.  15.  69,255.  16.  136,000. 
17.  5248.  18.  4644.  19.  1734.  20.  51,114.  21.  8676.  22.  3976. 
23.  4095.  24.  135,030. 

Page  44.  1.  24,442.  2.  46,748.  3.  45,904.  4.  66,825. 
5.  124,440.  6.  133,179.  7.  184,518.  8.  291,450.  9.  124,313. 
10.  112,420.  11.  207,739.  12.  379,080.  13.  50,061.  14.  197,290. 
15.  577,521.  16.  145,754. 

Page  45.  1.  106,889.   2.  23,150.   3.  16,535. 

Page  46.  1.  $3933.  2.  549,120ft.  3.  $498.96.  4.  26,708  Ib. 
5.  44,660  Ib.  6.  Gained  $1678.20. 


ANSWERS  401 

Page  48.    1.   621  A.      2.   54. 

Page  49.  3.  75,629.  4.  13.  5.  16  da.  6.  715  A.  7.  1105^ 
cd.  8.  16,107.  9.  66.  10.  Gained  $2500. 

Page  52.  l.  3,  3,  2,  2,  2,  2.  2.  2,  2,  31.  3.  11,  7,  3,  2,  2. 
4.  17,  17.  5.  5,  3,  3,  3.  6.  5,  3,  3,  3,  191.  7.  7,  5,  5,  3,  3. 
8.  3,  3,  7,  2,  2.  9.  3  and  317.  10.  3  and  509- 

Page  53.    i.   11.      2.   12.      3.   7.      4.   16. 

Page  54.     5.   12  ft.       6.   120  boards. 

Page  55.     1.  480.     2.   450.     3.   624.     4.  Jan.  1, 1904.    5.  252  A. 

Page  56.     1.   92f      2.   25. 

Page  57.  3.  15f .  4.  37f  5.  45  bu.  6.  5  bbl.  7.  720  yd. 
8.  5J  pc.  9.  8800  bu.  10.  3|  mi.  11.  5  sections.  12.  360  bbl. 
13.  500yd.  14.  $120.  15.  100  bbl. 

Page  62.    1.  $11,693,823.17.    2.  $1,417,548.52.    3.  $1,193,532.63. 

4.  $1,384,147.19.       5.    $24,320.65.       6.   $1088.60.       7.   $4073.76. 
8.    $384.51.       9.    $2094.15. 

Page  63.    1.   $209.11.     2.   $523.89.    3.   $326.33.    4.  $1256.81. 

5.  $5763.09. 

Page  64.  1.  Wheat,  350  bu.;  oats,  425  bu.;  corn,  175  bu. 
2.  $91.10. 

Page  68.  1.  71,116,542.  2.  1,436,942,736.  3.  13,644,817,552. 
4.  $4265.91.  5.  $299.64.  1.  195,448.  2.  $2081.48. 

Page  69.  3.  $1387.  4.  10  yr.  5.  6,075,486.  6.  20  bbl. 
7.  $7500;  $900.  8.  $  4500  and  $  4745.  9.  $540.19.  10.  240  bbl. 
11.  $4960.  12.  1,450,950,624.  13.  2,046,757,518. 

Page  73.     1.  4f .     2.  IJA.     3.  Ajyyu..    4.  *f|i.    5.  *£.    6.  AJA. 

7.    L3Q..         a    A^L.         9.    6|9.         10.    JJJLL.        11.    AfA.        12.    6ioi 
1.    142TV       2.    17^.       3.    27f|.       4.    13f|.       5.   38^.       6.   8^. 

7.  24fV.      8.    53||.      9.   23Jf      10.    16ff 

Page  74.    l.   f.    2.  ^    *  if-    *•  A*    5-  rVA-    6   «•    7-  f 

8.  }.      9.    |.     10.    A. 

Page  76.    1.  TV%,  A2ir,  AV    2.  IAA,  f||,  fff .     3.  «&  ffi,  ||5. 

*•  **»  li  **•    5-  *  *>  ^    6-  Wt»  itti  At     7-  tfc  ff»  *f 

8-  «b  *fc  Pi- 
page 77.     1.    Iff     2.    lA|f     3.    If     4.    1^     5.    If.      6.    Iff 

*    m-     8,    2^.     9.    IJAf     10.    Iff.     11.    HO..     12.    !40. 
Page  79..     1.   34,120.    2.  22,578f.    3.   12,934ff     4.   12,355|. 


402  ANSWERS 

Page  81.      l.   151&.      2.   T^,      3.   7ff.       4.   lO^fe.       5.  20. 

6.  llfi      7.   969f     8.   29J,     9.   25ff.     10.   47fV  acres. 

Page  84.      1.   25.      2.   7^.     3.  44.      4.   30.     5.   248.     6.   26f 

7.  462f.       8.    941.       9.   1^.       10.   11        11.   $  84|.       12.    $.63}. 
13.   $43f.      14.    $11.      15.    $200.      16.    274- 

Page  86.      l.   26811      2.   505|.      3.  339f.     4.   679}.      5.   756}. 

6.  316|.      7.   5421.      8.   6351       9.   90}.      IQ.   1533f.      11.   1831f. 
12.   939}. 

Page  89.    1.   $72.      2.   5  shares.       3.   40  families.       4.    28 1  bu. 

5.  5  da.     6.  $41     7.1  A.     8.  782|  sacks.     9.  18  da.     10.  13  fields. 

I.  2£f.     2.  7if     3.    $14,000.     4.   3bu.     5.   2  Ib.     6.    $135,000. 
Page  90.     7.   $52,500.     8.   $70.     9.   $300.     10.   $2.    11.   $5. 

12.  336  trees.        13.   A,  $  6,  B,  $  15,  and  0,  $  16.        14.   $35,  $40. 

15.  1050  horses.     16.   6}  bu.     17.   221  da.     ia   C.  $38f,  J.  $47f 
19.   114fft.     20.    Colt,  $94;  cow,  $30. 

Page  91.  21.  25|ft.  22.  405  sheep.  23.  Carriage,  $324; 
horse,  $  216.  24.  6|  da.  25.  5T\5T  da.  26.  42f  da.  27.  A,  $  1260 ; 
B,  $420;  C,  $840.  28.  Gained  $  10,106.56.  29.  A,  $2800; 

B,  $3500.  30.  2161.  31.  1200.  32.  240  bu.  33.  20  da. 

Page  92.  34.  $2501TV  35.  28  da.  36.  15  hr.  37.  A,  $19.75; 
B,  $15.80.  38.  L6se52^.  39.  60  oranges.  40.  28  bu.  and  80  bu. 
41.  $900.  42.  255}  bu.  43.  $11.91  gain.  44.  Gained  $38f 
45.  $438f 

Page  93.  46.  Gained  $  98.32.  47.  $395.54;  $9.17.  48.  Gained 
$3.14.  49.  $3277.40.  50.  $25,250. 

Page  96.    1.  .26.     2.  .27.     3.  .0006.     4.  .04.    5.  5.7.    6.  500.05. 

7.  .00022.       8.   5000.005.       9.   1,000,000.000001.        10.   .500  or  .5. 

II.  .00005. 

Page  97.     12.   7.7.      13.   2.002.      14.   2000.002.      15.   11.000107. 

16.  83.0504.  17.   710.00243.  18.   54,054,054.0054054054. 
19.   .37,  .0004,  1.097,  3.0893,  9.17. 

Page  98.    1.  ff    2.   ^    3.  ^.    4.   {fo    5.  Jffa    6. 

7-    iW&-  8.    IMfa       9.    «.       10.    flHHfr  11.    jfffa.     12. 

13.  ^.  14.    ytV        15.    ^.        16.    5^.  17.    13^-      18-    11} 
19.    31^.  20.    16^.       21.    81&£       22.  G&ftftp       23.   35^. 
24.   15^.  25.   28J.      1.   .0625.      2.   .15.  3.   .275.      4.   .09375. 
5.   .1375.  6.    .5652173  +  .         7.   .0525.  8.   .46875.        9.   .024. 
10.   .9375.  11.   .015625.         12.   .96875.  13.   .028.        14.   .95. 
15.   .94. 


ANSWERS  403 

Page  99.    i.   848.1816.         2.   1652.461772.         3.   12,638.517852. 

4.  1,000,608.012354001.        5.   57,697.358230005.        6.   385.8225yd. 
7.   41.885  cd.      8.   136.33  thousand  feet.      9.   84.423  T.      10.   Num- 
ber of  thousand  feet,  101.184;   total  cost,  $1403.75.  11.   376. 
12.   9262or926f 

Page  100!    l.   .52977.       2.  1.27848.       3.   5.5264.      4.  1.546548. 

5.  .81.       6.   .198.        7.   754.6005.       8.   .3148.       9.   385,994.01246. 
10.   1000.0099.      11.   2.99985.      12.   102.93702. 

Page  101.    l.  0.     2.   117.843385.     3.   .2375.     4.   .6.     5.   .0009. 

6.  7231.98325125.         7.   .0018044.         8.   $554.63.         9.   $336.33. 

10.  $14,856.56.       11.   $3510.71.       12.   $496.05. 
Page  102.    13.   $  1068.57.      14.   $  3537.58. 

Page  105.    1.   10,011,112.1010001.  2.   6,330,303.3333. 

3.  40,448,404.48.  4.   1,056,000.605.  5.   5,655,500.1005. 

6.  30,003.36303603. 

Page  106.     1.   2.92125.      2.  751.383957246471.      3.   $127.12. 
Page  107.     4.   6.875  da.        5.   24.93-f  thousand  feet.        6.   7  yr. 

7.  Gained  $5513.80.     8.  Gained  $  400.75.     9.  $1124.34.     10.  48  da. 

11.  Entire  school,  1000  pupils ;  bookkeeping  department,  500 ;  short- 
hand and  typewriting  department,  375;   English  department,  125. 

12.  $2238.75. 

Page  108.  13.   Total  gain,  $  5950 ;  net  gain,  $  5000.     14.  $7500. 

Page  112.  l.  371  Ib.       2.   81.5  yd. 

Page  113.  3.   115.2  A.       4.   689yd.       5.   123  Ib.       6.    $40.96. 

Page  115.  1.   $  2166.      2.   $  5163.97.      3.   $  1142.33. 

Page  117.  1.  $2643.52.  2.   $6315.38.  3.   $2462.43. 

4.  $8789.88.       5.   $8256.38. 

Page  118.  1.  $20;  $19.13.  2.  $28.35;  $28.  3.  $30.17; 
$8.  4.  $28.13;  $60.  5.  $121.50;  $16.  6.  $630.63;  $180. 
7.  $31.04;  $62.50.  8.  $11.17;  $12.  9.  $148.28;  $247.50. 
10.  $61.40;  $125. 

Page  119.  1.  $8.25.  2.  $14.66.  3.  $158.76.  4.  $123.18. 
>  $53.  6.  $38.75.  7.  $19.80.  8.  $100.32.  9.  $140.81. 
10.  $399.  1.  $2.68.  2.  $4.71.  3.  $78.70.  4.  $1.42. 

5.  $1985.26.     6.   $1609.78.     7.   $903.96.     8.   $21.06.     9.   $12.65. 
10.   $19.51. 

Page  120.  1.  $32.  2.  $29.71.  3.  $52.52.  4.  $39.29. 
5.  $24.66.  6.  $22.63.  7.  $55.67.  8.  $58.33.  9.  $174.44 
10.  $132.36. 


404  ANSWERS 


9. 

Page  128. 
Page  129. 
Page  130. 
Page  131. 
Page  132. 
$  97.44. 
Page  133. 

l. 

2. 
3. 
1. 
5. 

10. 

9  11,607.65. 
$  322.67. 
$  452.57.     4. 
$458.33.     2. 
$429.16.     6. 

(a)  $170.29 

$540.26. 
$524.03.    3.  $168.68. 
$943.54.     7.   $987.11. 

;    (6)  $136.23.          11. 

4. 
.8. 

(a) 

$  1040.65. 
$440.48. 

$381.70; 

(6)  $305.37. 

12.  (a)  10.  20's,  1;  10's,  9;  5>s,  7;  2>s,  9;  1's,  4;  halves,  4;  quar- 
ters, 3 ;  dimes,  3 ;  nickels,  2 ;  pennies,  14.  (b)  10.  10's,  9 ;  5's,  4 ; 
2's,  9;  1's,  4;  halves,  4;  quarters,  6;  dimes,  5;  nickels,  4;  pennies,  3. 
(a)  11.  20's,  11 ;  10's,  7 ;  5's,  11 ;  2's,  13 ;  1's,  5;  halves,  6;  quarters,  6 ; 
dimes,  7;  nickels,  7;  pennies,  15.  (b)  11.  20's,  8;  10's,  7;  5's,  8;  2's, 
11 ;  1's,  5 ;  halves,  9 ;  quarters,  9 ;  dimes,  11 ;  nickels,  7 ;  pennies,  17. 

Page  151.      1.   193,555m.      2.   12,363d.     3.   709  pt.     4.   175  qt. 

5.  155,243".    6.  561  gi.    7.  180,002  oz.    8.  238,475  cu.  in. 
9.  6,860,715  sq.  in.   10.  9543  1.   11.  296,065  oz.  12.  1373  pwt. 
13.  3767  in.       14.  3859J  sq.  ft.       15.  182,727.75  sq.  ft. 
16.  4,345,531  sec.   17.  194  cu.  ft.   la  40,396  gr.   19.  269  pt. 
20.  129,832m. 

Page  152.  1.  7  wk.  1  da.  15  hr.  20  min ;  or  1  mo.  20  da.  15  hr. 
20min.  2.  24  bbl.  20  gal.  1  qt.  3.  112  bu.  2  pk.  5  qt.  4.  8  A. 
66  sq.  rd.  3  sq.  yd.  4  sq.  ft.  72  sq.  in.  5.  3  mi.  124  rd.  2  yd.  8  in. 

6.  1  T.  17  cwt.  95  Ib.     7.  9  Ib.  1  oz.  5  pwt.  20  gr.    8.  73  yr. 
3  mo.  1  wk.  1  da.    9.  2  hr.  38  min.  57  sec.   10.  66  A.  72  sq.  rd. 
11.  5  cu.  ft.  152  cu.  in.  12.  17  T.  8  cwt.  32  Ib. 

Page  153.  1.  432  gr.  2.  10  pwt.  3.  213  rd.  1  yd.  2  ft.  6  in. 
4.  110  sq.  rd.  5.  112  A.  40  sq.  rd.  29  sq.  yd.  51.84  sq.  in.  6.  43  sq.  rd. 
19  sq.  yd.  2  sq.  ft.  36  sq.  in. 

Page  154.  1.  £^.   2.  A^fa.   3.  .12  T.   4.  .017361+  yd. 

Page  155.  1.  .327  T.  2.  .0625  A.  3.  fff  Ib.  4.  .29791+  Ib. 

5.  im  T. 

Page  156.  1.  £90  3s.  2.  17  mi.  46  rd.  1  yd.  2  ft.  3.  33  A. 
2  sq.  rd.  17  sq.  ft.  7  sq.  in.  4.  672  Ib.  1  oz.  12  pwt.  8  gr.  5.  211  Ib. 
11  oz.  18  pwt.  21  gr. 

Page  157.  1.  14  gal.  2  gi.  2.  396  A.  78  sq.  rd.  3.  3  yd.  2  ft. 
IJin.  4.  6  Ib.  9  oz.  7  pwt.  17  gr.  5.  $44.31.  6.  6  T.  5  cwt. 

7.  Gained  £507  5s. 


ANSWERS  405 

Page  159.  1.  96  da.  2.  236  da.  3.  323  da.  4.  87  da. 
5.  223  da.  6.  233  da.  7.  49  da.  8.  241  da.  9.  81  da. 
10.  402  da.  11.  523  da.  12.  883  da.  13.  5  yr.  7  mo.  3  da. 
14.  17  yr.  15.  4  yr.  4  mo.  5  da.  16.  2  yr.  10  mo.  25  da. 

17.  3  yr.  1  mo.  3  da.  18.  7  yr.  4  ino.  23  da.  19.  3  yr.  1  mo.  1  da. 
20.  18  yr.  8  da.  1.  $1924.39.  2.  $1958.25. 

Page  160.     3.   54  T.  16  cwt.  47  Ib. ;    $699.  4.   $70.20. 

5.  $577.78;  $650;  $433.33.        6.    $20,589.84.        7.    $97.50  gain. 

8.  $73.88.     9.   $  122.80  by  first  method ;  $122.79  by  second  method. 
Page  161.     l.   2bu.  3pk.        2.   1  T.  57  Ib.  12  oz.         3.   £4  15s. 

3f  far.       4.   3  yr.  46  da.  7  hr.  30  min.        5.   1  Ib.  3  oz.  5  pwt.  11  gr. 

6.  $13,050.  7.    $19.97.          8.   $5.          9.    £  1061  3s.  9d.  1  far. 
10.   £38  3d  2.4  far.;  £513  14s.  3d.  3.36  far.     1.   12.     2.   £770  11s. 
5d.  3  far.        3.  4  A.  83  sq.  rd.  6  sq.  yd.  644  sq.  in.        4.    101  sq.  rd. 
2  sq.  yd.  21.6  sq.  in.       5.    123  mi.  162  rd.  3  yd.  1  ft.  4  in. 

Page  162.     6.   $6.20  gain.         7.   Gained  3^.         8.   $  72.58  gain. 

9.  $19.16.          10.   $54.          11.    $72.81.          12.    Gained  $360.67. 
13.   $1209.60  gain.        14.   $9.77.        15.   $88.89. 

.  Page  164.    l.   75  ft.  4.7808  in.  2.   $  335.  3.   $  17.33. 

4.   440.6094ft.       5.    1250ft.       6.   168  ft.  4  in. 

Page  166.      1.   4  A.  116  sq.  rd.      2.   19  A.  122  sq.  rd.      3.   34  A. 

4.  10  A.  56  sq.  rd.     5.  15  A.     6.  12  A.  12  sq.  rd.     7.  446  A.  5  sq.  ch., 
or  446  A.  80  sq.  rd.         8.    249  A.  9  sq.  ch.,  or  249  A.  144  sq.  rd. 
9.  390  A.     10.  $3277.97.     11.  357  rd.  5  ft.  6  in.    12.  414  ft.  lOf  in. 
13.    Ark.,  177  A.  80  sq.  rd. ;   N.  and  S.  Dak.,  124  A.  40  sq.  rd. ;   all 
other  states,  133  A.  20  sq.  rd.  14.   $  10.56.  15.    $  121,500. 
16.    57  sq.  rd.  21  sq.  yd.  108  sq.  in.     17.    $100.     18.    $262.35. 

Page  168.  1.   63  yd.  2.    $  15.63.  3.   128£  yd. ;  130  yd. ; 

$327.25.     4.  65|yd.     5.    $150.     6.    61|  yd. ;  82J  yd. 

Page  169.  7.   $9.46.    8.    $13.10. 

Page  170.  1.   75  rolls.       2.    $8.75.       3.   14  rolls.       4.    $15.20. 

5.  $39.60. 

Page  171.  1.  $23.99.  2.  $23.11.  3.  $45.32.  4.  $77.61. 
5.  $38.85.  6.  $24.17. 

Page  172.  1.  $182.25.  2.  $63.  3.  $49.75.  4.  60,000  shingles. 
5.  48  M. ;  $  168.  6.  $  192. 

Page  174.      1.   $37.70.      2.   1458  cu.  ft.     3.  $284.28.     4. 
cu.ft.    5.   1570|cu.  ft.     6.36ft.     7.   $16.29. 


406  ANSWERS 

Page  176.  l.  420  board  ft.  2.  $14.04.  3.  $11.20.  4.  27  £ 
5.  $108.39.  6.  $23.42.  7.  $24.18.  8.  $21.78.  9.  $132. 
10.  $132.44. 

Page  178.  1.  39i|  cd.  2.  26  ft.  9&  in.  3.  $227.11.  4.  64ft. 
5.  $45.  1.  $6000.  2.  With  asphalt ;  save  $45,883.20.  3.  With 
brick;  save  $85.33.  4.  147,840  blocks.  5.  $58,594.44. 

Page  180.    1.   345.6  bu.     2.   1040  bu.     3.   768  bu.      4.   1920  bu. 

5.  480  bu.        6.   1536  bu.        7.   237.476+  bbl. 

Page  181.     8.  64  rd.  9.6+  ft.    9.  1903.5648  gal.    10.  22.3074  gal. 

Page  182.  1.  81^-  or  81.45+  perches.  2.  118  cu.  yd.  14  cu.  ft., 
or  118^4  cu.  yd.  3.  836.1  perches.  4.  $1701.68.  5.  284,460 
bricks ;  280,104  bricks.  6.  $283.50.  7.  546,920  bricks ;  535,040 
bricks.  1.  $8100.  2.  8ft.  3.  $7200.  4.  $32.58. 

Page  183.  5.  $25.34.  6.  $75.92.  7.  $57.60.  8.  $120. 
9.  $140.  10.  90  pupils.  11.  $21.36.  12.  $4937.50.  13.  $5. 
14.  149J  cu.  ft. ;  1792  board  ft. ;  -jfccu.ft.  15.  $3465.  16.  Cheaper 
to  pave  with  asphalt;  $10,260.80.  17.  $165. 

Page  188.  1.  $60,000.  2.  $2700.  3.  Wheat,  8320  bu. ;  oats, 
3120  bu. ;  barley,  13,520  bu. 

Page  189.  4.  $17,820.  5.  $14.04.  6.  $4.40.  7.  $576.60; 
658  bu.  8.  $40,500. 

Page  190.    l.  50%.    2.  25%.     3.  95%.     4.  5f%.     5.   54ii%. 

6.  800%.      7.   20%.      8.    100%.      9.   50%.      10.   33^%. 
Page  191.     1.  24.      2.  5%.      3.   $1900. 

Page  192.  4.  2040  qt.  5.  $40,300.  6.  $3750.  7.  200  trees. 
8.  $10.56.  9.  $54.  10.  370f  bu. 

Page  193.  1.  $600.     2.  $400.     3.  $20,000.    4.  $4.50.     5.  $28. 

6.  500  pupils.     7.  36,080.     8.  $  8000.     9.  560  bu.     10.  200,100. 
Page  194.     1.   $10,000.      2.  $20,840.      3.   $3200. 

Page  195.  4.  Horse,  $  300 ;  carriage,  $  156.  5.  Monday,  $500; 
Tuesday,  $  400 ;  Wednesday,  $600.  6.  $32.40.  7.  $175.  8.  A, 
$1000;  B,  $600.  1.  23f£%;  42«%;  33|%.  2.  $300.  3.  $1959.38. 
4.  A,  $7125;  B,  $2500.  5.  $7200;  $12,000.  6.  1£%  gain. 

7.  $13,650.       8.   331%. 

Page  196.  9.  122,048.  10.  40  gal.  11.  27¥V%;  21|^%; 
18TfT%.  12.  25%.  13.  Amount  of  indebtedness,  $  2252.08 ;  in 
bank,  $9008.33.  14.  Lost,  $666.67.  15.  $23,400.  16.  $64,000 
17.  $60,250. 


ANSWERS  407 

Page  197.  18.  Carriage,  $150;  horse,  $187.50;  harness,  $50. 
19.  Cost,  $308;  lost,  $28.  20.  ¥^%;  .000125;  3642%.  21.  5  cwt. 
61.48  lb.,  or  5  cwt.  61  Ib.  7.68  oz.  22.  First  cost,  $7480.50; 

8r9°847%-  23-  Wife>  $21,760*  daughter,  $10,000;  younger  son, 

$12,500;  elder  son,  $13,750.  24.  $1750;  $3062.50;  $6125;  and 
$8575.  25.  $3850.  26.  $20,000.  27.  133%%.  28.  Grazing, 
504  A. ;  grain,  420  A. ;  timber,  936  A. 

Page  198.    29.  $7200.    30.   50%. 

Page  200.  1.  $826.67.  2.  $567.  3.  $2400.  4.  $252.80. 
5.  360yd.  6.  $72.  7.  First;  $|.  8.  Gained  $18. 

Page  201.   9.   66f%;  $420.     10.   $374.75. 

Page  202.  1.  $680.24.  2.  $1246.80.  3.  $322.87.  4.  Amount 
of  bill  to  render,  $4863.75;  amount  to  be  remitted,  $4620.56. 
5.  Amount  of  bill  to  render,  $2199;  amount  to  be  remitted, 
$2133.03. 

Page  203.  6.  $222.30;  trade  discount,  $91;  cash  discount, 
$11.70.  7.  $234.38.  8.  $270. 

Page  205.  l.  12%.  2.  $1000.  3.  $165.75.  4.  $920. 
5.  $1000.  6.  2|%. 

Page  206.  1.  Lead  pipe,  35^;  iron  pipe,  13^;  bath  tubs,  $8.64, 
$7.20,  and  $5.76.  2.  $1317.50.  3.  $126.  4.  Second  is  $  9  cheaper. 
5.  $189.47. 

Page  207.     1.  Gain,  $  24.75 ;  selling  price,  $  173.25.     2.  $  6406.25. 

3.  Gained,  $74.     4.    $105  gain.     5.    $37.50  gain. 

Page  208.  1.  2000  bu.  2.  29^%.  3.  49J%.  4.  6J%. 
5.  331%.  6.  50%.  7.  77J%. 

Page  209.    8.   29^%.         1.   $5000.         2.   $2750.         3.   $40. 

4.  $7085.71.      5.    $150. 

Page  210.  1.  $10,500.  2.  $  60  and  $  100.  3.  $1000.  1.  $2000. 
2.  22f%.  3.  $1218.75.  4.  400  bbl. 

Page  211.  5.  1000  bu.  6.  25%.  7.  16|%.  8.  $20,010. 
9.  22%  loss.  10.  $625.  11.  $11.25.  12.  37|%.  13.  $2. 
14.  46f%.  15.  $31f.  16.  331%.  17.  $74.25. 

Page  212.  18.  $9350.  19.  16|%  loss.  20.  $10,000.  21.  $110. 
22.  $16.80.  23.  $59,320.  24.  3f%  loss.  25.  15^.  26.  18J|% 
loss.  27.  $58.60. 

Page  213.     28.   9|f  gal.     29.   14f  %  gain.     30.   $300.     31.   30  £ 
32.    50^.          33.    20%.          34.    491%.          35.    23l2V%-         36- 
37.    $1080;  2f%.       38.    $250. 


408  ANSWERS 


Page  214.    39 
Page  215.     1. 
$.ST.   $EH. 

,    50%.     40. 

d}>   Tj1    Ti    A                    < 

«jp  _CJ._LW\_         < 

141.  4 
$E.IA 

l.   $5985 

2. 

Z.OA      , 

.      42.    $6. 

$I.EA 

fl 

AG 

BW' 

1    l 

i 

;i. 

PI.BW 

EO  $ 

$T.MB' 

.    $IER. 

$  M.LB 
$  ELUA 

$.TH' 

$HA. 

LB 

*'$ 

M 

.LC'  $ 

M.BS 

*'  $  SHB' 

$  IO.TB 

'  $.TH 

<H!   C!/"\  *2J  "C1  T  A 

A  tlP     O-V/  «I1)    -ElaXJlL 


'   $TH'   $I.WB 

Page  217.    1.   $1.67.    2.   94^.     3.   $2.43.     4.   $2.72.     5. 
6.    $3.39.     7.    74^.     8.    $2.53.     9.    $1.75.     10.    $2.10.     11.    15  £ 
12.   $1.33.      13.   $3.75.      14.    $10.88.      15.    $3.55.       16.   $2.75. 
17.   $2.67.       18.   $6.      1.   $40.       2.    $12. 

Page  218.     3.   $64.      4.   25%.      5.   8J%.      1.   $7.      2.   Shoes, 

$5;    boots,    $4;    rubbers,    85^.  3.    Hosiery,    25^;    26^;    40^; 

50^.     Knit    goods,    75^;    43^;    $1.70;    $1.08.  4.    Hosiery, 

$.AY.     $.AH.     $.NA.     $.DY        ,,-   ..          ,      $  .OY.     $.ND. 

'        ~'    -'  '  >  '  ' 


$  B.OE  '    $ 

•                       **.        J.J.O.UO,      <ff  U.^JJJ  ,        giW  CO,      «J£>    J-.-l-L^       lllCO,      <J\J  f  , 

6.   Hats,    $1 

.67;    $1.17;    $2.78;    $3.89.       Gloves,   $2;    $2.33; 

$  1.50.       7. 

9£       8.   64^.     Page  219.     9.  26%  gain.      10.  284^. 

Page  221. 

1.    Comm.,    $225;    remitted   to   principal,    $11,025. 

2.   $93.75. 

3.   $1110.       4.   $3937.50.       1.   2J%.       2.   2%. 

Page  222. 

1.   $  90,000.       2.   50  bales. 

Page  223. 

3.   9000  bu.       4.    $16,910.40. 

Page  224. 

1.   30,000  Ib.     2.  145  doz.     3.   4000  Ib.  ;  comm.,  $  30. 

4.   1750  Ib. 

Page  225. 

1.  $2929.70.     2.  $4818.26.     3.  $394.48. 

Page  226. 

4.  $740.88.     5.   $1896.30.     6.    $67.50.      7.   225  bbl. 

a  4|-%.     9. 

33£%.     10.    2000  Ib.  ;$  16.20  comm.     11.    $1818.60. 

Page  227. 

12.  $1097.40.      13.  $1888.       14.  1890  bbl.;  $1.11. 

15.  $1508.57  comm.;  $16,091.43.    16.  5|^.    17.  10,666f  yd.   18.  5%. 

19.  Insurance,  \%;  comm.,  2%;  net  proceeds,  $  12,453.    20.  14,400  bu. 
Page  230.    1.    $9.13.          2.   $2.66.          3.   $3.68.          4.   $4.68. 

5.   $6.44.     6.    $2.62.     7.    $7.55.     8.   $11.10.     9.   $9.05.     10.  77^. 
11.   58  £       12.    $1.99.       13.    $10.19.       14.    $15.24.       15.    $4.99. 

16.  $8.59.          17.    $5.84.          18.    $15.03.          19.    $1.36;  $5.27. 

20.  $11.08;  $16.04.      21.    $34.99;  $49.81.      22.    $81.92;  $74.36. 
23.    $68.06;  $32.58.     24.    $24.40;  $34.     25.    $12.04;  $37. 


ANSWERS 


409 


Page  232.  1.  9091.  2.  $1713.79.  3.  $2774.75.  4.  $2408. 
5.  $1758.17.  6.  $4298.  7.  $2995.54.  8.  $1788.98.  9.  $2453.03. 
10.  $1877.98.  11.  $2429.89.  12.  $7515.75.  13.  $2405.70. 
14.  $1851.64.  15.  $2433.23.  16.  $1708.28. 


Page  235.  l. 

Page  236.  1. 

Page  237.  4. 

Page  238.  1. 

6.   $3.05.  7. 

li:  $14.72.  12. 

16.   $4.28.  17.   $2.80. 

Page  239.  1.   $4.38. 


$41.95.     2.   $60.62.     3.   $110.15.     4.   $352.88. 
$461.40.    2.   $169.63.     3.   $110.58. 
$  157.67. 

$7.95.     2.   $2.25.    3.  $21.74.    4.  $.20.     5.  $.32. 
$2.20.         8.   $1.51.         9.    $5.58.        10.    $4.35. 
$4.70.       13.   $1.57.      14.  $2.01.      15.   $1.83. 
18.   $.88. 

2.    $5.25.         3.    $3.71.         4.    $17.44. 


5.   $9.24.     6.    $1.13.     7.   $5.83.     8.  $3.33.     9.  $1.76.     10.  $5.70. 


11. 
16. 
21. 
26. 
31. 


$2.34. 
$1.15. 
$6.61. 
$2.39. 

$7.79. 


13.   $6.32. 
18.    $11.65. 
23.   $25.BO. 
28.    $8.22. 


14.   $3. 
19.    $1.34. 
24.    $10r60. 

29.   $6.18. 


15.    $24.58. 

20.   $10.96. 

25.    $25.38. 

30.    $2.82. 


34.   $14.30. 

35.    $1.41 

36.    $1.47. 

39.    $10.50.      40.    $7.70. 

41.   $79.20. 

2.    $7.84. 

3.    $2.13. 

4.    $11.67. 

¥8.    a  $1.11. 

9.  $17.70. 

10.    $12.67. 

12.    $8.02. 
17.   $4.37. 
22.   $19.71. 
27.    $4.86. 
32.    $1.T9. 

Page  240.     33.   $8.78. 
37.    $.42.      38.    $12.87. 
42.   $29.96.    43.   $80.84. 
Page  242.    1.   $19.75. 
5.    $52.     6.    $28.80.     7. 
11.   $38.53.    12.   $8.80. 

Page  243.     1.  $221.40.    2.  $2800.53.    3.   $300.28.     4.   $305.97. 

5.    $1640.63.     6.   $1785.     7.   $4981.67.     8.   $98.90.     9.   $100.83. 

Page  244.     1O.   $2766.55.      1.   $11.53.      2.   $5.18.      3.   $3.62. 

4.  $2.80.         5.   $5.19.         6.    $16.81.         7.    $3.95.         8.    $5.10. 
9.   $16.95.     10.    $73.97.     11.   $15.28.     12.   $25.70.     13.   $68.24. 
14.   $21.57.     15.   $21.37. 

Page  245.    1.  4%.     2.  61%.    3.   1\%.    4. 5%. 
Page  246.    1.   2  yr.  5  mo.  24  da.  2.   3  yr,  .10  mo.  12  da. 

3.   April  25,  1881. 
Page  247.    1. 
Page  248.    1. 
Page  249.    1. 
Page  250.    1. 

5.  $176.12. 


$4408.16.    2.   $3204.23.    3.   $27,500. 
$3000.     2.   $4000.     3.   $5000.    4.   $2000. 
$202.08.     2.   $21.60.     3.   $366.60.     4.  $2422.30. 
$372.%.      2.   $96.44.      3.   $68.66.     4.   $588,46. 


410  ANSWERS 

Page  253.    1.   $2921.83.  2.   $2024.80.  3.   $38.64. 

4.  $253.23.         5.   $2766.76. 

Page  254.     1.   $1291.71.     2.  4%.     3.   6%;  $600.     4.   $51.20. 

5.  4%. 

Page  255.  6.  27  da.  7.  10%;  10  yr.  8.  $74.82.  9.  $23.61. 
10.  $537.97.  11.  12%.  12.  $920.08.  13.  $20.72.  14.  \\%. 
15.  3  yr.  7  mo.  24  da.  16.  1\  yr.  17.  6£i%. 

Page  256.    18.   $4644.61.    19.   May  18,  1905.    20.   $2728.82. 

Page  257.     1.   Interest;  $12.29. 

Page  258.     2.   Better  to  pay  cash ;  55  £  3.   No  difference. 

4.  Lose  $11.82.     5.   $1523.15.     6.  $7.48.     7.   $6.40.     8.  4%  loss. 
9.   $15,000.          10.   $4000.          11.   $29.79.         12.   July  1,  1903. 

13.  $10,855.79. 

Page  259.    14.  4%.    15.   $9736.94. 

Page  266.  1.  Apr.  15;  80  da.  2.  Apr.  7;  23  da.  3.  July  6; 
64  da.  4.  May  20;  49  da.  5.  Dec.  20;  91  da.  6.  Aug.  4;  47  da. 
7.  Sept.  7;  23  da.  8.  Jan.  1 ;  73  da.  9.  Feb.  22;  9  da.  10.  Nov.  6; 
25  da.  11.  Feb.  27 ;  60  da.  12.  Mar.  1 ;  84  da.  13.  June  1 ;  85  da. 

14.  Sept.  29;  81  da.     15.    Feb.  14;  28  da.     16.    Mar.  14;  26  da. 
Page  267.     1.    Bank    discount,    $17.60;       proceeds,    $2382.40. 

2.    $9.80;  $1190.20.     3.   $18.28;  $3231.72.      4.   $3.13;  $496.87. 

5.  $22.08;  $2477.92.     6.  $36;  $3564.     7.  $15;  $1485.  8.  $32; 
$2368.     9.   $13.50;  $4486.50.     10.   $13.87;  $2386.13.  11.   Pro- 
ceeds, $  1188. 

$  1200.00.  (Your  place),  Apr.  18, 19—. 

Sixty  days  after  date  we  promise  to  pay  to  the  NATIONAL  BANK 
OF  REDEMPTION,  or  order,  Twelve  Hundred  Dollars,  value  received. 

J.  M.  Cox  &  Co. 


Page  268.     12.   Maturity,  July  14,  1903 ;  term  of  discount,  74  da. ; 
discount,  $11.10;  proceeds,  $1188.90.     13.   June  1,1903;  60  da.; 


ANSWERS  411 

$45;  $4455.  14.  Feb.  25,  1904;  55  da.;  $11;  $889.  15.  Aug.  3, 
1903 ;  63  da. ;  $  5.87  ;  $  664.94.  16.  Feb.  29, 1904  ;  177  da. ;  $97.55; 
$2382.45.  17.  Feb.  28,  1903;  18  da. ;  $2.41;  $800.92. 

Page  269.  18.  July  5, 1903 ;  65  da. ;  $  9.95 ;  $  1214.05.  19.  Aug. 
23,1903;  85  da. ;  $36.50;  $2539.75.  20.  Oct.  16,  1903;  56  da. ; 
$1.54;  $218.46.  21.  Nov.  5,1903;  35  da. ;  $2.46;  $419.04. 

Page  270.  22.  July  14, 1904;  54 da.;  $7.16;  $787.54.  23.  Bank 
discount,  $10;  proceeds,  $1190.  24.  $3483.28.  25.  Maturity, 
May  19, 1903 ;  bank  discount,  $  7.20 ;  proceeds,  $  1792.80.  26.  Over- 
drawn, $  123.98.  27.  $420.48,  to  their  credit.  28.  $2.43;  $500.07. 
29.  $2.35;  $647.76. 

Page  271.  1.  $659.98.  2.  $600.  3.  $4000.  4.  $2400. 
5.  $2400.  6.  $355.64. 

Page  274.  1.  $550.40.  2.  $845.69.  3.  $1661.87.  4.  $344.45. 
5.  $3089.10.  6.  $1181.24.  f^tf>  ^o  ^ 

Page  276.  1.  $635.92.  2.  $122.29.  3.  $279.64.  4.  At  6%, 
$183.49;  at  5%,  $179.70;  at  8%,  $191.08.  5.  $180.93. 

Page  282.  i.  Oct.  18, 1903.  2.  Nov.  15, 1903.  3.  June  23, 1903. 
4.  Dec.  23,  1903. 

Page  284.    1.   Apr.  11, 1903.    2.   June  5,  1903.     3.   July  6,  1903. 

4.  Jan.  24,  1904.  5.   May  17,  1903.  6.   Nov.  18,  1903. 
7.   July  22,  1903. 

Page  288.    l.   Dec.  9,  1902. 

Page  289.     2.  Feb.  6,  1903.    3.  Apr.  18, 1903.    4.  Sept.  10, 1903. 

5.  Aug.  8,  1903. 

Page  290.     1.   Dec.  7,  1903. 

Page  291.     2.   Dec.  31, 1903. 

Page  293.  1.  $313.18.  2.  $613.78.  3.  Equated  date,  Jan.  4, 
1902  ;  cash  balance,  $435.15. 

Page  295.     1.   $2034.71.      2.  $1730.26. 

Page  296.    3.   $1985.?3. 

Page  298.     1.   $752.50. 

Page  299.    2.   $781.04.      1.   $10. 

Page  300.     2.    $1264.74. 

Page  304.  l.  $3637.50.  2.  $7275.  3.  $17,381.25.  4.  $8820. 
5.  $26,400.  6.  $25,967.50. 

Page  305.  7.  $  92,868.75.  8.  $  2917.50.  9.  $  80,500,  par 

value;  $  83,263.75,  market  value.  10.  110.  l.  400  shares; 

$  40,000.  2.  500  shares.  3.  740  shares.  4.  300  shares ;  $  1200. 


412  ANSWERS 

Page  306.  1.  $64,837.50.  2.  $29,718.75.  3.  $180. 
4.  $51,700. 

Page  307.  1.  5.268+  %.  2.  N.Y.  Air  Brake,  better  by 

1.481-1-%.  3.  4.593+%.  1.  $1440.  2.  $75,475.50. 

Page  308.  3.  Total  dividend,  $9,892,460;  C's  dividend,  $110. 
4.  Total  dividend,  $251,250;  F's  dividend,  $93.75.  5.  Surplus 
fund,  $47,085.79;  dividend,  $800,000;  undivided  profits,  $498,- 
222.46.  1.  9%.  2.  6  %  dividend ;$  11,135.60,  undivided  profits. 

Page  309.     3.  13%;  $  567,274.83  undivided  profits.    1.  $1352.06. 
2.   $415.38  loss. 
Page  310.    3.    $3782.55. 
Page  312.     1.   4.347+%.  2.   $3600.62.  3.   $62,625. 

4.  3.897+%.          5.   $  63,825  invested ;     4.23  +  %,  rate   of  income. 
6.    $57.50.     7.    50  bonds.     8.  44  bonds;  3.936+  %,  rate  of  income. 

Page  313.  9.  Mo.  Pacific,  better  by  .741 +%.  l.  $81,940.63. 
2.  $1000.  3.  $72,031.25.  4.  Quotation,  121 J ;  brokerage,  $  62.50. 

5.  5.024+%.       6.   3.636+%.      7.    $  1004.05  to  make  margin  good ; 
$  1079.05  loss  if   sold   out.       8.   127  bonds ;   unexpended  balance, 
$485.62. 

Page  314.  9.  $  36.25  gain;  $3343.75  on  hand.  10.  Am.  Ice  Sees., 
8.139+  %  ;  Del.  &  Hud.,  4.176+  %  ;  Bait.  &  Ohio,  5.068+  %  ;  Erie, 
2d  pf.,  6.324+  %.  11.  Surplus  fund,  $508,067.90;  dividend, 

$  3,500,000 ;     undivided  profits,   $  6,153,290.10.      12.    $  50,437.50. 
13.    41%.  14.    Dividend,  10  %;    undivided  profits,  $  955,946. 

15.    Y's  rate   of  income,    5.298+  %;  Z's  5.263+  %;   Y's  dividend, 
$  3000  ;  Z's,  $  2500. 

Page  318.    1.   $117.     2.   $840. 

Page  319.  1.  1J  %•  2-  I  %-  l-  $4450.  2.475.  3.  $37,500. 
1.  $300.  2.  $8000.  3.  $1576.50. 

Page  320.  4.  $  523.75.  5.  f  %.  6.  $27,411.17.  7.  $  11,454.55. 
8.  $18,125.  9.  $8.66.  10.  j  %.  11.  $1967.96.  12.  $60. 

13.  G.,  $  630;    H.,  $  150;  and  M.,  $  337.50  ;  $  142.50  gain;  .71  %. 

14.  H.,  $26,522.73;     M.,  $43,099.43;     A.,  $23,207.39;     Phoenix, 
$  26,522.73 ;  Provident,  $  26,522.73. 

Paee321.    15.  $  47,500  and  $  39,375. 

Page  325.      1.  $158.10.     2.  $280.40.    3.  $333.40.     4.  $6386.91 


ANSWERS  413 

due  from  bank;  $1386.94  more  from  bank  than  from  insurance 
company.  5.  $4610.64  more  than  premiums;  $3947.08  more  than 
the  premiums  and  interest.  6.  $  680.05  loan ;  $  1555.70  due  from 
bank.  7.  $  6948  on  ordinary  policy ;  $  24,885.60  on  endowment 
policy.  8.  $4265.92  more  than  premiums;  $3985.33  loss  on  policy. 
9.  $29.60  less  in  premiums,  per  $1000;  $1276.08  in  principal  and 
interest,  per  $1000. 

Page  326.  10.  $28,026.25.  11.  $4011.05  more  than  premiums; 
$3891.27  less  than  amount  received  by  A.  12.  $440.55,  loan  value; 
9  years^47  days,  extended  insurance;  $925,  amount  of  paid  up 
policy.  13.  $  2596.60,  loan  value ;  13  years,  extended  insurance  ; 
$  3710,  paid  up  policy. 

Page  328.    l.  $18.75.      2.  $1134;  B.  $13.44. 

Page  329.  l.  Rate,  \\%\  $120.  2.  50^  per  $100;  $11.25. 
1.  Bate,  £%;  $62.50.  2.  $3550.  3.  4.5  mills ;  $70.  4.  $35.35. 

5.  Kate,  .012898 ;  $  82.39. 

Page  330.  6.  21  mills;  2£  mills.  7.  4.5  mills;  $110.63. 

8.  Kate,  5|  mills;  $54.05.  9.  1.25669 £  10.  $24,519.60; 

$612.99. 

Page  333.  1.  $24,300.  2.  $7273.56  total  tax;  $10,910.34 

if  he  failed  to  make  return.  3.  $76,647,  total  city  tax;  $790, 
total  suburban  tax;  $2182.50,  total  farm  tax;  $44.25,  tax  horses 
and  cattle ;  $  26,743.10,  total  tax  money  at  interest;  $  5.60  tax  ve- 
hicles to  hire ;  grand  total,  $106,412.45.  4.  $6687.18.  5.  $66. 

6.  $94.48.  7.   $243,  discount;     $24,057,  total  paid  for  real 
estate  tax.     8.   $4.74. 

Page  337.     1.    $1235.60.  2.   $610.50.  3.   $933.33. 

4.  $1238.64.        5.   40  gal. 

Page  338.     l.  $1178.80.  2.  $5645.50.   3.  $2010.  4.  $27,031.80. 

5.  $10,656.40.         1.    $866.80.         2.    $576.30.          3.   $1317.60. 

Page  339.  4.  $3644.55.  5.  $155.90.  6.  $2494.  7.  $463.20. 
8.  $1168.80. 

Page  340.  9.    $210.08.          10.   $1.80.        11.   $16.82. 

Page  344.  l.   $848.87.  2.   $1393.     3.   $784. 

Page  345.  4.  $5000.  5.  $8696.16.  6.  $1378.30.  7.  $18/1.30, 
8.  $1890.50.  9.  $5080.98;  $344.42  gain.  10.  $2471.88. 


414  ANSWERS 

Page  352.  1.  $1946.  2.  11,160.89  francs,  or  11,160  francs  89 
centimes.  3.  $5103.48.  4.  £1195  17s.  6d.  5.  25,000  marks. 
6.  $1666.87.  7.  1864  marks  8.  $25,876.26.  9.  5.185  francs. 
10.  $.40f. 

Page  353.     11.   $2186.21.     12.   $10,558.96. 

Page  355.     1.    $  2003.72.  2.   $3790.30.      ,      3.   $1501.87. 

4.  $3506.93.        5.   $3056.63.       6.   $3594.11.       7.   Mk.  19,429.43. 
8.    $2827.12. 

Page  356.     9.   $2851.18.  10.   $119.37.  11.   $5824.17. 

12.   $82.15  gained.        13.  $8.16.        14.   £  15  14s.  6Jd.  more. 

Page  358.  1.  $10,600.  2.  A,  $2400;  B,  $1200;  C,  $800. 
3.  A,  $120;  B,  $300;  C,  $320.  4.  $12,000.  5.  $15,000. 
6.  A,  $150;  B,  $200;  C,  $300. 

Page  359.  7.  $64,800.  8.  A,  $3000;  B,  $6000;  C,  $9000; 
D,  $12,000.  9.  llfff  da.;  C,  ^$ 32.67;  H,  $25.41;  T,  $22.87; 
L,  $  19.05. 

Page  362.  1.  A,  $126;  B,  $168.  2.  A,  $6500;  B,  $6500; 

C,  $3250. 

Page  363.  3.  A's  share  of  gain,  $  13,750 ;  B's  share  of  gain, 
$8250;  A's  share  of  selling  price,  $26,250;  B's  share  of  selling 
price,  $15,750.  4.  A's  gain,  $398.88;  C's  gain,  $319.10;  B's  gain, 
$239.32;  A's  present  worth,  $  1648.88 ;  C's  present  worth,  $1319.10; 
B's  present  worth,  $989.32.  5.  B,  $6000;  C,  $4000;  D,  $2000. 
6.  Net  gain,  $795.51 ;  A's  share  of  net  gain,  $530.34;  B's  share  of 
net  gain,  $265.17;  A's  present  worth,  $5530.34;  B's  present  worth, 
$2665.17.  7.  Net  loss,  $488.50;  C's  present  worth,  $3755.75; 
D's  present  worth,  $1755.75.  8.  Net  gain,  $2255;  E's  present 
worth,  $8405.66;  Fs  present  worth,  $8751.67;  G's  present  worth, 
$7751.67.  9.  $7290. 

Page  365.    1.   A,  $276.92;  B,  $346.15;  C,  $276.93. 

Page  366.     2.    A,  $596.53;  B,  $559.25;  C,  994.22.  3.  A, 

$17.40;  B,  $29.70;  C,  $48.     4.   A,  $36;  B,  $32;  C,  $20;  D,  $4. 

5.  M's   gain,  $1737.93;   E's  gain,  $1862.07;  M's  present  worth, 
$3737.93;  E's  present  worth,  $6362.07.  6.   A,  $4548.39;  B, 
$2951.61.      1.   A's  present  worth,  $12,025.17;  B's  present  worth, 
$9525.18;    firm's    present    worth,    $21,550.35;    firm's    net    gain, 
$4050  35. 


ANSWERS  415 

Page  368.  2.  Net  resources,  $23,564.25;  net  solvency,  $23,564. 25 1 
net  gain,  $1064.25.  3.  Burke,  $17;533.33;  Brace,  $  17,83-3.34; 
Baldwin,  $  17,633.33.  4.  Loss,  $22,747.09;  Brigg's  loss,  $5907.62; 
Parson's  loss,  $16,839.47;  net  insolvency,  $2187.09;  Brigg's  present 
worth,  $127.38;  Parson's  insolvency,  $2314.47. 

Page  369.  5.  A's  in  vestment,  $3865.80;  B's  in  vestment,  $2577.20; 
firm's  insolvency,  $5557;  A's  insolvency,  $3334.20;  B's  insolvency, 
$  2222.80.  6.  Net  gain,  $  2636.83 ;  Mason's  present  worth,  $  5818.42; 
Eiver's  present  worth,  $5818.41.  7.  Net  gain,  $2865.14;  D's 
present  worth,  $8632.57;  E's  present  worth,  $8432.57. 

Page  370.  8.  $2100.  9.  Present  worth  of  business,  $  16,766.57 ; 
net  gain,  $4673.09 ;  A's  present  worth,  $8371.78 ;  L's  present  worth, 
$8394.79.  10.  Gain  of  each,  $1825;  E.  H.  Hill's  present  worth, 
$7425;  N.  P.  Pond's  present  worth,  $7825. 

Page  373.     1.   $3200.25.        2.   $1910.40.        3.   $1565. 

Page  374.  1.  7th  meeting,  dues,  $550;  interest,  $16.50;  loan, 
$550;  profit  per  share,  $.03;  book  value  of  each  share,  $7.105. 
2.  13th  meeting,  dues,  $400;  interest,  $26.40;  loan,  $400;  profit 
per  share,  $  .066 ;  book  value  of  each  share,  $  13.429. 

Page  375.  3.  17th  meeting,  dues,  $800;  interest,  $  96 ;  loan, 
$1600;  profit  per  share,  $.12;  book  value  of  each  share,  $18.02. 
25th  meeting,  dues,  $  800 ;  interest,  $156;  loan,  $800;  profit  per 
share,  $.195;  book  value  of  each  share,  $27.3175.  4.  $1  profit 
for  24th  month,  or  $12.16  total  profit  for  24  months;  $212.16,  book 
value  after  paying  25th  month's  dues.  5.  $  18.40.  6.  $  22. 

Page  376.  1.  L's  profit,  $475.20;  M's,  $206.25;  N's,  $158.40; 
O's,  $110;  P's,  $  16.50.  Lowest  terms,  respectively,  36,  25,  16, 4, 
and  1.  Profit  per  share,  respectively,  $3.96,  $2.75,  $1.76,  $.44, 
and  $.11.  2.  S's  profit,  $83.60;  T's,  $37.62;  Q's,  $104.50; 
U's,  $12.54;  V's,  $9.50;  E's,  $14.25.  Lowest  terms,  for  S,  16; 
for  T,  9 ;  for  Q,  U,  and  V,  4 ;  for  R,  1.  Profit  per  share,  respect- 
ively, $  1.52,  $  .855,  $  .38,  $  .095.  3.  $  274.70. 

Page  379.  1.  733J  ft.  2.  478  bu.  3.  $2812.50.  4.  2  yr.  9  mo, 
23  da.  (221  da.).  5.  140  A. 

Page  380.  1.  $567.  2.  350  rd.  3.  26f  acres.  4.  $384.75. 
5.  93  da. 


416 


ANSWERS 


Page  382.  l.  $40.    2.    $64.    3.   $64.50. 

Page  383.  i.  $211.    2.  36.09. 

Page  384.  3.  $483.96.    4.    $368.48.     5.    $321.80. 

Page  389.  l.  143,814.44  francs.      2.    229.2784  Ib. 

120  sq.  rd.  4.  $6431.23        5.   $29.81.       6.   11.69. 


3.   617  A. 

7.   $1.32. 


8.   6£.      9.   3i  meters;  11  ft.  5.79  in.      10.    1535.2722  kilometers. 
11.    54  meters;  59.0544yd.     12.    2J  hectares;  5.55975  A. 

Page  392.  i.  14.  2.  15.  3.  24.  4.  35.  5.  75.  6.  206. 
7.  125.  8.  1024.  9.  11.2.  10.  7.09.  11.  f|.  12.  ff 

Page  393.  i.  100  ft.  2.  480  rd.  3.  273.137  +  rd.  4.  452  rd. 
9  ft.  .56  -f  in.  5.  77.88  +  ft.  6.  3339.364  ft.  7.  100  rd.  8.  208.710  ft. 

Page  397.  l.  12.  2.  25.  3.  48.  4.  404.  5.  f  6.  H- 
7.  £1.  8.  10.813  +  .  9.  ff  l.  2067.954  +  sq.  in.  2.  15ft.  3.  10ft. 
9  -h  in.  4.  7  f  t.  7  +  in.  deep  j  15  ft.  2  +  in.  square.  5.  13  ft.  10  +  in. 


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